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Full text of "Studies in deductive logic"

STUDIES IN 
DEDUCTIVE LOGIC 



a 



Tl c 



B 




HAG GNANTIAI OVDIC 




TIC YneNANTIAI OVH 




STUDIES 



IN 



DEDUCTIVE LOGIC 

SL ^anual for >tutient0 



BY 



W. STANLEY JEVONS 

LL.D. (EDINB.)) M.A. (LOND.), F.R.S. 



SECOND EDITION 



3Lonton 

MACMILLAN AND CO. 
1884 

The right of translation and reproduction is reserved. 




ELECTRONIC VERSION 
AVAILABLE 



Printed by R. & R. CLARK, Edinburgh. 



PREFACE. 



IN preparing these Studies I have tried to carry forward 
the chief purpose of my Elementary Lessons in Logic, which 
purpose was the promotion of practical training in Logic. 
In the preface to those Lessons I said in 1870: The 
relations of propositions and the forms of argument present 
as precise a subject of instruction and as vigorous an exer 
cise of thought, as the properties of geometrical figures or 
the rules of Algebra. Yet every schoolboy is made to learn 
mathematical problems which he will never employ in after 
life, and is left in total ignorance of those simple principles 
and forms of reasoning which will enter into the thoughts of 
every hour. ... In my own classes I have constantly 
found that the working and solution of logical questions, 
the examination of arguments and the detection of fallacies, 
is a not less practicable and useful exercise of mind than 
is the performance of calculations and the solution of 
problems in a mathematical class. 

The considerable use which has been made of the 
Elementary Lessons seems to show that they meet an educa 
tional want of the present day. The time has now perhaps 



viii PREFACE. 

arrived when facilities for a more thorough course of logical 
training may be offered to teachers and students. 

For a long time back there have been published books 
containing abundance of mathematical exercises, and not 
a few works consist exclusively of such exercises. In 
recent years the teachers of other branches of science, 
such as Chemistry and the Theory of Heat, have been 
furnished with similar collections of problems and numerical 
examples. There can be no doubt about the value of such 
exercises when they can be had. The great point in 
education is to throw the mind of the learner into an active, 
instead of a passive state. It is of no use to listen to a 
lecture or to read a lesson unless the mind appropriates 
and digests the ideas and principles put before it. The 
working of problems and the answering of definite questions 
is the best, if not almost the only, means of ensuring this 
active exercise of thought. It is possible that at Cambridge 
mathematical gymnastics have been pushed to an extreme, 
the study of the principles and philosophy of Mathematics 
being almost forgotten in the race to solve the greatest 
possible number of the most difficult problems in the 
shortest possible time. But there can be no manner of 
doubt that from the simple addition sums of the schoolboy 
up to problems in the Calculus of Variations and the 
Theory of Probability, the real study of Mathematics must 
consist in the student cracking his own nuts, and gaming 
for himself the kernel of understanding. 

So it must be in Logic. Students of Logic must have 
logical nuts to crack. Opinions may differ, indeed, as to 



PREFACE. ix 

the value of logical training in any form. That value is 
twofold, arising both from the general training of the mental 
powers and from the command of reasoning processes 
eventually acquired. I maintain that in both ways Logic, 
when properly taught, need not fear comparison with the 
Mathematics, and in the second point of view Logic is 
decidedly superior to the sciences of quantity. Many 
students acquire a wonderful facility in integrating differen 
tial equations, and cracking other hard mathematical nuts, 
who will never need to solve an equation again, after they 
settle down in the conveyancer s chambers or the vicar s 
parsonage. With the ordinary forms of logical inference 
and of logical combination they will ceaselessly deal for 
the rest of their lives ; yet for the knowledge of the forms 
and principles of reasoning they generally trust to the light 
of nature. 

I do not deny that a mind of first-rate ability has con 
siderable command of natural logic, which is often greatly 
improved by a severe course of mathematical study. But 
I have had abundant opportunities, both as a teacher and 
an examiner, of estimating the logical facility of minds of 
various training and capacity, and I have often been 
astonished at the way in which even well-trained students 
break down before a simple logical problem. A man who 
is very ready at integration begins to hesitate and flounder 
when he is asked such a simple question as the following : 
If all triangles are plane figures, what information, if any, 
does this proposition give us concerning things which are 
not triangles? As to untrained thinkers, they seldom 



x PREFACE. 

discriminate between the most widely distinct assertions. 
De Morgan has remarked in more than one place l that 
a beginner, when asked what follows from Every A is B, 
answers Every B is A of course The fact that such a 
converse is often true in geometry, although it cannot be 
inferred by pure logic, tends to mystify the student. Al 
though all mathematical reasoning must necessarily be 
logical if it be correct, yet the conditions of quantitative 
reasoning are often such as actually to mislead the reasoner 
who confuses them with the conditions of argumentation in 
ordinary life. A mathematical education requires, in short, 
to be corrected and completed, if indeed it should not be 
preceded, by a logical education. There was never a 
greater teacher of mathematics than De Morgan ; but from 
his earliest essay on the Study of Mathematics to his very 
latest writings, he always insisted upon the need of logical 
as well as purely mathematical training. This was the 
purpose of his tract of 1839, entitled, First Notions of Logic 
preparatory to the Study of Geometry, subsequently reprinted 
as the first chapter of the Formal Logic. A like idea 
inspired his valuable essays On the Method of Teaching 
Geometry, quoted above. 

1 The Schoolmaster : Essays on Practical Education, 1836, vol. ii. 
p. 1 20, note. This excellent essay On the Method of Teaching 
Geometry was originally printed in the Quarterly Journal of Education, 
No. XI. 1833, vol. vi. pp. 237-251. Similar views are put forth 
in De Morgan s earlier work, On the Stiidy and Difficulties of Mathe 
matics, published in 1831 by the Society for the Diffusion of Useful 
Knowledge. See chapter xiv. See also De Morgan s Fourth Memoir 
on the Syllogism, p. 4, in the Cambridge Philosophical Transactions 
for 1860. 



PREFACE. xi 

Professor Sylvester, indeed, in his most curious tractate 
upon the Laws of Verse (p. 19), has called in question the 
nut -bearing powers of logic, saying : It seems to me 
absurd to suppose that there exists in the science of pure 
logic anything that bears a resemblance to the infinitely 
developable and interminable euristic processes of mathe 
matical science. To such a remark this volume is 
perhaps the best possible answer, especially when it is 
stated that I have had great difficulty in selecting and 
compressing my materials so as to get them into a 
volume of moderate size. If any person who thinks with 
Professor Sylvester should object to the greater part of 
the problems as dealing with concrete logic, let him look 
to the end of this book, where he will find that the closely 
printed Logical Index to the forms of law governing the 
combinations of only three terms, fills four pages, without 
in any way including the almost infinitely various logical 
equivalents of those distinct forms. He will also learn that 
a similarly complete index of the forms of logical law 
governing the combinations of only five logical terms would 
fill a library of 65,536 volumes. Surely there is scope 
enough here for euristic processes. 

An anxious and difficult task which I had to encounter in 
compiling this book consisted in choosing the system or 
systems of logical notation and method which were to be 
expounded. When once the convenient but tyrannical 
uniformity of the Aristotelian logic was overthrown, each 
writer on the science proceeded to invent a new set of 



xii PREFACE. 

symbols. But it is impossible to employ alike the Greek 
letters of Archbishop Thomson, the mysterious spiculae 
of De Morgan, the cumbrous strokes, wedges, and dots of 
Sir W. Hamilton, and the intricate mathematical formulae 
of Boole. After a careful renewed study of the writings 
of these eminent logicians I felt compelled in the first 
place to discard the diverse and complicated notative 
methods of De Morgan. Few or none admire more than 
I do the extraordinary ingenuity, fertility, and, in a certain 
way, the accuracy of De Morgan s logical writings. My 
general indebtedness, both to those writings and to his 
own unrivalled oral teaching, cannot be sufficiently ac 
knowledged. I have, moreover, drawn many particular 
hints from his works too numerous to be specified. 
Nevertheless, to import his mysterious spiculae into 
this book was to add a needless stumbling-block. The 
question would have arisen too, which of his various 
systems to adopt; for De Morgan created six equally 
important concurrent syllogistic systems, the initial letters 
of the names of which he characteristically threw into the 
anagrams, Rue not! True? No! These systems were 
the Relative, Undecided, Exemplar, Numerical, Onymatic, 
and Transposed. See A Budget of Paradoxes, pp. 202-3. 
There was in fact an unfortunate want of power of general 
isation in De Morgan; his mind could dissect logical 
questions into their very atoms, but he could not put the 
particles of thought together again into a real system. As 
his great antagonist, Sir W. Hamilton, remarked, De 
Morgan was wanting in Architectonic Power. 



PREFACE. xiii 

It seems equally impossible, however, to adopt Sir 
W. Hamilton s own logical symbols. His chief method 
of notation has been briefly described in the Elementary 
Lessons in Logic (p. 189). He also constructed or con 
templated other systems of notation, as stated in his 
Lectures on Logic (vol. iv., pp. 464476). In no case 
do these notations seem to be so good as the earlier and 
simpler one of Mr. George Bentham. And after a 
laborious reinvestigation, rendered indispensable by the 
composition of various parts of this book, I have been 
forced to the conviction that in almost every case where 
Hamilton differed from contemporaries or predecessors 
he blundered. He was, as his admirers said, to put the 
keystone into the arch of the Aristotelic syllogism ; but, 
in spite of his Architectonic Power I fear we must 
allow that his arch has collapsed. (See pp. 129133, 
151-4, and 157-8, of this book.) 

With the logical innovations of Dr. Thomson the case 
is different. While he appears to enjoy the credit of an 
independent discovery of the Quantification of the 
Predicate, prior to any public and explicit statement of 
the same by Hamilton, De Morgan, or Boole, but 
posterior to the neglected work of Mr. George Bentham, 
he did not commit the blunders of Hamilton, nor overlay 
his work with useless crowds of short-hand symbols. He 
most aptly completed the ancient scholastic notation of 
propositions (A, E, I, O) by adding U, Y, ^ and w to 
denote the new forms derived from Quantification of the 
Predicate, carefully showing at the same time that ?? and 

b 



xiv PREFACE. 

w are practical nonentities. I have therefore used his 
notation for quantified propositions and syllogisms where 
necessary. 

Boole s great works are of course the foundation of almost 
all subsequent progress in formal logic. My own views, 
as I long since explicitly stated, 1 are moulded out of his. 
Believing, however, that the mathematical dress into which 
he threw his discoveries is not proper to them, and that 
his quasi -mathematical processes are vastly more compli 
cated than they need have been, I have of course preferred 
my simpler version. Students who wish to comprehend 
Boole s power and Boole s methods must go to the original 
writings. It is really impossible that any abstract or 
summary can give an adequate idea of the stupendous 
efforts which Boole made to construct a general mathe 
matical calculus of inference. Dr. Macfarlane, of Edinburgh, 
has lately published a new version of Boole s system 
under the title Algebra of Logic, but I am unable as 
yet to discover that he has made any improvement on 
Boole. 

The writings of M. Delboeuf on Algorithmic Logic, 
first printed in the Revue Philosophique for 1876, and since 
reprinted, are very interesting, but were written in ignorance 
of what had been done in this country by Boole and 
others. 

Quite recently Mr. Hugh MacColl, B.A., has published in 
the Proceedings of the London Mathematical Society, and in 
Mind, several papers upon a Calculus of Equivalent 
1 Pure Logic, 1864, p. 3, etc. 



PREFACE. xv 

Statements, which arose out of an earlier article in the 
Educational Times ^ His Calculus differs in several points 
both from that of Boole and from that described in this 
book as Equational Logic. Mr. MacColl rejects equations 
in favour of implications ; thus my A = AB becomes with 
him A : B, or A implies B. Even his letter-terms differ 
in meaning from mine, since his letters denote propositions, 
not things. Thus A : B asserts that the statement A 
implies the statement B, or that whenever A is true, B is 
also true. It is difficult to believe that there is any 
advantage in these innovations; certainly, in preferring 
implications to equations, Mr. MacColl ignores the necessity 
of the equation for the application of the Principle of 
Substitution. His proposals seem to me to tend towards 
throwing Formal Logic back into its ante-Boolian con 
fusion. 

In one point, no doubt, his notation is very elegant, 
namely, in using an accent as a sign of negation. A is 
the negative of A j and as this accent can be applied with 
the aid of brackets to terms of any degree of complexity, 
there may sometimes be convenience in using it. Thus 
(A + B) = A B ; (ABCD ...) = A + B + C + 
D + . . . . I shall occasionally take the liberty of using 
the accent in this way (see p. 199), but it is not often 
needed. In the case of single negative terms, I find ex 
perimentally that De Morgan s Italic negatives are the best. 
The Italic a is not only far more clearly distinguished from 
A than is A , but it is written with one pen-stroke less, 
1 August 1871, also July 1877. 



xvi PREFACE. 

which in the long run is a matter of importance. The 
student, of course, can use A for a whenever he finds it 
convenient. 

The logical investigations of Mr. A. J. Ellis, F.R.S., require 
notice, because they are closely analogous to, if not nearly 
identical with, my own. I am much indebted to him for 
assisting me to become acquainted with his views. Not 
only has he supplied me with an unpublished reprint, with 
additions, of his articles in the Educational Times, but he 
has allowed me access to the manuscripts of two elaborate 
memoirs which he presented to the Royal Society, and 
which are now preserved in the archives of the Society. 
Some account of these investigations will be found in the 
Proceedings of the Royal Society for April 1872, No. 134, 
vol. xx. p. 307, and November 1873, vol. xxi. p. 497. 
In the former place Mr. Ellis remarks : The above con 
tributions are believed to be entirely original .... Jevons 
first led my thoughts in this direction, but all resemblance 
between us is entirely superficial. The question of resem 
blance thus raised by Mr. Ellis must be left to others to 
decide; but in order to avoid possible misapprehension, 
I must say, that however different in symbolic expres 
sion, Mr. Ellis s logical system seems to me identical in 
principle with my own. The developments of the Com 
binational Method, as described in the Educational Times 
(June, July, and August, 1872), are substantially the same as 
I had previously published in several papers and books. 
Mr. Ellis also employs card diagrams of combinations 
arranged upon the ledges of a black-board, which practi- 



PREFACE. xvii 

cally form the Logical Abacus, as described by me in 
1869. 

The only point in which I am conscious of having 
received assistance from Mr. Ellis has regard to the 
necessary presence of combinations and the significance 
of their total disappearance as proving contradiction. 
I may not have sufficiently insisted upon the importance 
of this matter; but the fact is that so long ago as 1864 
(see pp. 1 8 1, 192, of this book) I pointed out the complete 
disappearance of ^a letter-term from the combinations as 
the criterion of contradiction in the conditions governing 
logical combinations, and the same principle is explicitly 
stated in the Principles of Science (1874, vol. i. p. 133; new 
edition, p. 116). In the latter part of this book I have 
more fully developed the theory of the relation of pro 
positions, often turning as it does upon this criterion of 
contradiction. This theory will, I think, be found to be 
the natural development of ideas stated in my earlier 
essays; but I may have received some hints from 
Mr. Ellis s writings. The above remarks apply only to 
such portions of Mr. Ellis s Memoirs as treat of logical 
combination and inference; other portions in which he 
investigates sequence in space and time, probability, 
etc., are not at all in question. 

The Logical Index, although now printed for the first 
time, has been in my possession since 1871 (see Principles 
of Science, ist edition, vol. i. pp. 157, 162; new edition, 
pp. 137, 141, etc.); but it is only by degrees that I have 
appreciated the wonderful power which it gives over all 



xviii PREFACE. 

logical questions involving three terms only; and it is quite 
recently that it has occurred to me how it might be printed 
in the form of a compact and convenient table. 

Mr. Venn has published in the Philosophical Magazine 
for July 1880, a paper On the Diagrammatic and 
Mechanical Representation of Propositions and Reason 
ings. An article on Symbolic Reasoning by the same 
author will also be found in Mind for the same month. 
The text of this book having been completed and placed 
in the printer s hands before Mr. Venn s ingenious papers 
were published, it has not been possible to illustrate or 
to criticise his views. 

I may mention that M. Louis Liard, Professor of Philo 
sophy at Bordeaux, who had previously explained and 
criticised the substitutional view of Logic in the Revue 
Philosophique (Mars, 1877, torn, iii., p. 277, etc.), has 
since published a very good though concise account of the 
principal recent logical writings in England, under the 
title, Les Logiciens anglais contemporains (Paris : Germer 
Bailliere, 1878). 

These Studies consist in great part of logical Questions 
and Problems gathered from many quarters. In the majority 
of cases I have indicated by initial letters the source or 
authorship of the questions when clearly known (see the 
List of References on p. xxv) ; but I have not always 
carried out this rule, and in not a few cases the questions 
have been printed several times already, and are of 
doubtful authorship. A large remaining fraction of the 
questions and problems are new, and have been de- 



PREFACE. xix 

vised specially for this book. As shown by the author s 
name appended, a few questions have been borrowed 
from the work of the Very Rev. Daniel Bagot, Dean 
of Dromore, entitled Explanatory Notes on the Principal 
Chapters of Murray s Logic . . . with an Appendix of 3 3 7 
Questions to Correspond. A few excellent illustrations 
have also been drawn from a privately printed tract on 
Logic by the late Sir J. H. Scourfield, M.P., his own 
annotated copy having been kindly presented to me by 
the author a few years before his death. 

In forming this compilation I have been more than ever 
struck by the fact that the larger part of logical difficulties 
and sophisms do not turn upon questions of formal logic 
but upon the relations which certain assertions bear to the 
presumed or actual knowledge of the assertor and the 
hearer. If the person X remarks that c All lawyers are 
honourable men, it is one question what is the pure 
logical force of this proposition, as measured by its 
effect on the combinations of the terms concerned and 
their negatives. It is quite another matter what X means 
by it ; why he asserts it ; what he expects Y to understand 
by it ; and what Y actually does take as the meaning 
of X. 

Under certain circumstances assertions convey a meaning 
the direct opposite of what they convey at other times. 
If a man is taken with a fit and the first medical man who 
arrives says, You must not think of putting the man 
under the pump/ the man will not be put under the 
pump ; but if the identically same assertion is made about 



xx PREFACE. 

the centre of interest of an excited and angry mob, the 
man goes to the pump. It is evident that there ought to 
exist a science of applied deductive logic, partly corre 
sponding to the ancient doctrines of rhetoric, in which 
the popular force of arguments as distinguished from their 
purely logical force should be carefully analysed. A few 
questions and answers given in this book may perhaps 
belong, properly speaking, to rhetorical logic (see pp. 119, 
140-1, etc.), but I have not found it practicable to pursue 
the subject in this book. It should be evident that a 
thorough comprehension of the purely logical aspect of 
assertions must precede any successful attempt to in 
vestigate their rhetorical aspect. I may possibly at some 
future time attack the problems of rhetorical logic. 

A further question which forced itself upon my notice 
was that of the practicability of including exercises in 
Inductive Logic. As Mr. H. S. Foxwell suggested, in 
ductive exercises and problems are even more needed 
than those of a deductive character. But, on consideration 
and trial, it seemed highly doubtful whether it would be 
possible to throw questions of inductive logic into the 
concise and definite form essential to a book of exercises. 
I have given abundance of inverse combinational problems 
which are really of an inductive character (see pp. 2528); 
but exercises in the inductive methods of the physical 
sciences, if practicable at all, would require a much greater 
space, and a very different mode of treatment from that 
which they could receive in this work. For the present, 
at all events, I must content myself with referring readers 



PREFACE. xxi 

to the ample exposition of inductive methods contained 
in the 3d, 4th, and 5th books of the Principles of 
Science. 

Some readers may perhaps be still inclined to object 
to the Syllogism, and to deductive logic generally, that it 
is comparatively worthless, because all new truths are 
obtained by induction. This doctrine has prevailed with 
many writers from the time of John Locke to that of John 
Stuart Mill. But if I have proved in Chapters VI., VII., 
XL, XII., and in other parts of the Principles of Science, 
that induction is the inverse operation of deduction, the 
supreme importance of syllogistic and other deductive 
reasoning is not so much restored as explained. In 
reality the cavillers against the syllogism have never suc 
ceeded in the slightest degree in weakening the hold 
of the syllogism upon the human mind : it was against 
the nature of things that they should succeed. Their 
position was as sensible as that of a tutor who should 
recommend his pupils to begin Mathematics with Compound 
Division, but on no account to trouble themselves with the 
obsolete formula of the Multiplication Table. In every 
point of view, then, a thorough command of deductive 
processes is the necessary starting-point for any attempt to 
master more difficult and apparently more important pro 
cesses of reasoning. 

In the composition of the didactic parts of this book, 
I have tried the experiment of throwing my remarks into 
the form of answers to assumed, or in many cases actual, 
examination questions. I cannot call to mind any book 



xxii PREFACE. 

in which this mode of treatment has been previously 
adopted, but it seems to lend itself very readily to the 
clear exposition of knotty points and difficulties. In spite 
of much popular clamour against examinations, I maintain 
that to give a clear, concise, and complete written answer to 
a definite question or problem is not only the best exercise 
of mind, but also the best test of ability and training, 
which can be generally applied. 

The Frontispiece contains rough facsimiles of ancient 
logical diagrams which I copied from the fine MS. of 
Aristotle s Organon in the Ambrosian Library at Milan 
(L. 93, Superior). During a visit to Italy in 1874, I was 
much surprised and interested by the multitudes of curious 
diagrammatic exercises to be found in the logical MSS. 
of the great public libraries of Italy. The abundance 
of these diagrams shows that rudimentary logical exer 
cises were very popular in the country where, and at the 
time when, the dawn of modern science began to break. I 
estimated that a single MS. in the Biblioteca Communale 
at Perugia (Aristotelis de Interpretatiom cum Comment. 
A, 55. Grsece. Chart. 1485) contained at least eight 
hundred such diagrams. Those given in the frontis 
piece are the most ancient which I could discover. The 
MS. containing these (among others) is assigned in the 
printed catalogue to the eleventh or twelfth century, but 
the librarian was of opinion that it might belong to the 
tenth century. The figure in the centre shows the Greek 
original of the familiar Square of Logical Opposition, which 
has survived to this day (see p. 31). The triangular and 



PREFACE. xxiii 

lunular figures represent respectively the syllogistic moods 
Darapti, and (I believe) Datisi. 

To the imperfect list of the most recent writings on 
Symbolical Logic, given in this preface, I am enabled to 
add at the last moment the important new memoir of 
Professor C. S. Peirce on the Algebra of Logic, the first part 
of which is printed in the American Journal of Mathematics, 
vol. iii. (i5th September 1880). Professor Peirce adopts 
the relation of inclusion, instead of that of equation, as the 
basis of his system. 

BRANCH HILL, 
HAMPSTEAD HEATH, N.W., 
3^ October 1880. 



NOTE TO SECOND EDITION. 

THE present Edition has been printed from the Author s 
own copy, in which he had marked the few corrections 
and alterations which have now been made. 

HARRIET A. JEVONS. 



REFERENCE LIST 

OF INITIAL LETTERS SHOWING THE AUTHORSHIP 
OR SOURCE OF QUESTIONS AND PROBLEMS. 

A = PROFESSOR ROBERT ADAMSON, Owens College, 
Manchester. 

B = PROFESSOR ALEXANDER BAIN, University of Aberdeen. 

C = Cambridge University. Moral Science Tripos, or 
College Examination Papers. 

D = Dublin University. 

E = Edinburgh University. PROFESSOR ERASER. 

H = REV. JOHN HOPPUS, formerly Professor of Logic, etc. 
in University College, London. 

I = India Civil Service Examinations. 

L = London University, Second B.A., Second B.Sc., M.A., 
M.D. and D.Sc. Examinations. 

M = PROFESSOR THOMAS MOFFET, President and Professor 
in Queen s College, Galway. 

O = Oxford University. 

P = PROFESSOR PARK, Queen s College, Belfast, and 
Queen s former University. 

R = PROFESSOR GROOM ROBERTSON, University College, 
London. 

W = WHATELY S Elements of Logic. 



CONTENTS. 

CHAPTER pAGE 

I. THE DOCTRINE OF TERMS I 

II. QUESTIONS AND EXERCISES RELATING TO TERMS . 9 

III. KINDS OF PROPOSITIONS !8 

!V. EXERCISES IN THE DISCRIMINATION OF PROPO 
SITIONS 25 

V. CONVERSION OF PROPOSITIONS, AND IMMEDIATE 

INFERENCE 31 

VI. EXERCISES ON PROPOSITIONS AND IMMEDIATE 

INFERENCE ^ 

VII. DEFINITION AND DIVISION 64 

VIII. SYLLOGISM t , l 

!X. QUESTIONS AND EXERCISES ON THE SYLLOGISM . 94 
X. TECHNICAL EXERCISES IN THE SYLLOGISM . .103 

XI. .CUNYNGHAME S SYLLOGISTIC CARDS . . . 107 

XII. FORMAL AND MATERIAL TRUTH AND FALSITY . m 
XIII. EXERCISES REGARDING FORMAL AND MATERIAL 

TRUTH AND FALSITY I22 

XIV. PROPOSITIONS AND SYLLOGISMS IN INTENSION . 126 

XV. QUESTIONS ON INTENSION ^ 



xxviii CONTENTS. 

CHAPTER PAGE 

XVI. HYPOTHETICAL, DILEMMATIC, AND OTHER KINDS 

OF ARGUMENTS . . 137 

XVII. EXERCISES IN HYPOTHETICAL ARGUMENTS . . 145 
XVIII. THE QUANTIFICATION OF THE PREDICATE . .149 
XIX. EXERCISES ON THE QUANTIFICATION OF THE 

PREDICATE . 159 

XX. EXAMPLES OF ARGUMENTS AND FALLACIES . .164 

XXI. ELEMENTS OF EQUATIONAL LOGIC . 179 
XXII. ON THE RELATIONS OF PROPOSITIONS INVOLVING 

THREE OR MORE TERMS . . 223 

XXIII. EXERCISES IN EQUATIONAL LOGIC . . 227 

XXIV. THE MEASURE OF LOGICAL FORCE . . 249 

XXV. INDUCTIVE OR INVERSE LOGICAL PROBLEMS . 252 

XXVI. ELEMENTS OF NUMERICAL LOGIC . . 259 

XXVII. PROBLEMS IN NUMERICAL LOGIC 276 

XXVIII. THE LOGICAL INDEX 281 

XXIX. MISCELLANEOUS QUESTIONS AND PROBLEMS . 290 



STUDIES IN 

DEDUCTIVE LOGIC. 



CHAPTER I. 

THE DOCTRINE OF TERMS. 

INTRODUCTION. 

1. IN accordance with custom, I begin this book of 
logical studies with the treatment of Terms. Besides being 
customary, this way of beginning is convenient, because 
some difficulties which might otherwise be encountered in 
the treatment of propositions and arguments are cleared 
out of the way. But the continued study of logic convinces 
me that this doctrine of terms is really a composite and for 
the most part extra-logical body of doctrine. It is in fact 
a survival, derived from the voluminous controversies of the 
schoolmen. 

2. The difficulties of metaphysics, of physics, of grammar, 
and of logic itself, are entangled together in this part of 
logical doctrine. Thus, if we take such a term as colour ; 
and endeavour to decide upon its logical characters, we 
should say that it is categorematic, because it can stand as 
the subject of a proposition ; it is positive, because it im- 

B 



2 DOCTRINE OF TERMS. [CHAP. 

plies the presence rather than the absence of qualities. But 
is it abstract or concrete ? If concrete, it should be the 
name of a thing, not of the attributes of a thing. Now 
colour is certainly an attribute of gold or vermilion ; never 
theless, colour has the attribute of being yellow or red or 
blue. Thus I should say that yellowness is an attribute of 
colour, and if so, colour is concrete compared with yellow 
ness or blueness, while it is abstract compared with gold or 
cobalt. If this view is right, abstractness becomes a question 
of degree. 

3. Again, a relative term is one which cannot be thought 
except in relation to something else, the correlative. Thus 
nephew cannot be thought but as the nephew of an uncle 
or aunt; an instrument cannot be thought but as the 
instrument to some end or operation. But the question 
arises, Can anything be thought except as in relation to 
something else ? What is the meaning of a table but as 
that on which dinner is put ? What is a chair but the seat 
of some person ? Every planet is related to the sun, and 
the sun to the planets. Even meteoric stones moving 
through empty space are related by gravity to the sun 
attracting them. All is relative, both in nature and 
philosophy. 

4. As to the distinctions of general, singular, and proper 
terms, connotative and non-connotative terms, etc., they 
seem to me to be involved in complete confusion. I have 
shown in the Elementary Lessons in Logic (pp. 4 J -44) tnat 
Proper Names are certainly connotative. There would be 
an impossible breach of continuity in supposing that, after 
narrowing the extension of thing successively down to 
animal, vertebrate, mammalian, man, Englishman, educated 
at Cambridge, mathematician, great logician, and so forth, 
thus increasing the intension all the time, the single re- 



i.] DEFINITIONS AND EXAMPLES. 3 

maining step of adding Augustus de Morgan could remove 
all the connotation, instead of increasing it to the utmost 
point. But however this and many other questions in the 
doctrine of terms may be decided, it is quite clear in any 
case that this part of logic is ill-suited for furnishing good 
exercises in reasoning. This ground alone is sufficient to 
excuse rny passing somewhat rapidly and perfunctorily over 
the first part of logic, and going on at once to the subject 
of Propositions which offers a wide field for useful exercises. 
Accordingly, after giving brief definitions of the several 
kinds of terms, a few answers to questions, and a fair 
supply of unanswered questions and problems, I pass on to 
the more satisfactory and prolific parts of logic. 



DEFINITIONS AND EXAMPLES. 

5. A general term is one which can be affirmed, in the 
same sense, of any one of many (i.e. two or more) things. 

Examples Building, .front-door, lake, steam-engine. 

6. A singular term is one which can only be affirmed, in 
the same sense, of one single thing. 

Examples Queen Victoria, Cleopatra s Needle, the 
Yellowstone Park. 

7. A collective term is one which can be affirmed of 
two or more things taken together, but which cannot be 
affirmed of those things regarded separately or distributively. 

Examples Regiment, century, pair of boots, baker s 
dozen, book (a collection of sheets of paper). 



4 DOCTRINE OF TERMS. [CHAP. 

8. A concrete term is a term which stands for a thing. 

Examples Stone, red thing, brute, man, table, book, 
father, reason. 

9. An abstract term is a term which stands for an 
attribute of a thing. 

Examples Stoniness, redness, brutality, humanity, 
tabularity, paternity, rationality. 

10. A connotative term is one which denotes a subject 
and implies an attribute. 

Examples Member of Parliament denotes Gladstone, 
Sir Stafford Northcote, or any other individual 
member of parliament, and implies that they can 
sit in parliament.; bird denotes a hawk, or eagle, 
or finch, or canary, and implies that they have all 
the attributes of birds. 

11. A non-connotative term is one which signifies an 
attribute only, or (if such can be) a subject only. 

Examples Whiteness denotes whiteness only, an 
attribute without a subject. John Smith (according 
to J. S. Mill, and some other logicians) denotes a 
subject or person only, without implying attributes. 

12. Concrete general names are always connotative. 
Such also are all adjectives, without exception. Every 
adjective is the name of a thing to which it is added, and 
implies that the thing possesses qualities. Red is the name 
of blood or of other red thing, and implies that it is red. 
Redness is the abstract term, the name of the quality 
redness. 



i.] DEFINITIONS AND EXAMPLES. 5 

13. A positive concrete term is applied to a thing in 
respect of its possession of certain attributes ; a positive 
abstract term denotes certain attributes. 

Examples Useful, active, paper, rock; usefulness, 
activity, rockiness. 

14. A negative term is applied to a thing in respect of 
the absence of certain attributes ; if abstract the term 
denotes the absence of such attributes. 

Examples Useless, inactive, not-paper ; uselessness, 
inactivity. 

15. An absolute term is the name of a thing regarded 
per se, or without relation to anything else, if such there 
can be. 

Examples Air, book, space, water. 

1 6. A relative term is the name of a thing regarded in 
connection with some other thing. 

Examples Father, ruler, subject, equal, cause, effect. 

17. A categorematic term is one which can stand alone 
as the subject of a proposition. 

Examples Any noun substantive ; any adjective, any 
phrase or any proposition used substantively. 

1 8. A syncategorematic term is any word which can only 
stand as the subject of a proposition in company with some 
other words. 

Examples Any preposition, conjunction, adjective used 
adjectively. 

19. Differences of opinion may arise concerning almost 
every one of the definitions given above, and it would not 



6 DOCTRINE OF TERMS. [CHAP. 

be suitable to the purpose of this book to discuss the matter 
further. 

In every case, too, we ought before treating any terms 
to ascertain clearly that there is no ambiguity about their 
meanings. An ambiguous term is not one term, but two or 
more terms confused together, and we should single out one 
definite sense before we endeavour to assign the logical 
characteristics. The ambiguity of terms has however been 
sufficiently dwelt upon in the Elementary Lessons, Nos. iv. 
and vi., and it need not be pursued here. 

For the further study of the subject of terms the reader 
is referred to the Elementary Lessons ; Mill s System of Logic, 
book i., chapters i. and ii. ; Shedden s Logic, chapters i. 
and ii. ; Levi Hedge s Logic, part ii., chapter i. ; Martineau, 
Prospective Review, vol. xxix., pp. 133, etc. ; Hamilton s 
Lectures on Logic, vol. iii., lectures viii. to xii. ; Woolley s 
Introduction to Logic, part i., chapter i. 



QUESTIONS AND ANSWERS. 

20. Describe the logical characters of the follow 
ing terms Equal, equation, equality, equalness, 
inequality, and equalisation. 

Equal is a noun-adjective ; concrete, as denoting equal 
things ; connotative, as connoting the attribute of equality ; 
general, positive, relative; and syncategorematic, because 
it cannot as an adjective form the subject of a proposition. 

Equation, noun-substantive, originally abstract, as mean 
ing either equality, or the action of making equal. It is 
now generally used by mathematicians to denote a pair of 



i.] DEFINITIONS AND EXAMPLES. 7 

quantities affirmed to be equal. It is thus concrete, general, 
positive, perhaps absolute, and categorematic. 

Inequality is a noun-substantive, abstract, singular, nega 
tive, and categorematic. 

Equalisation means the action of making equal, an attri 
bute or circumstance of things, not a thing. It is thus 
abstract, singular, positive, categorematic. 



21. What are the logical characters of the terms, 
drop of oil, oily, oiliness ? 

A drop of oil being a concrete, finite thing, its name will 
be concrete, general, positive, relative (as having dropped 
from a mass of oil), collective as regards the particles of 
oil, connotative as implying the qualities of oiliness, etc., 
and categorematic. 

Oil is concrete, positive, collective, connotative and cate 
gorematic, like drop of oil, and only differs in not admitting, 
as regards any one kind of oil, of the plural. It is a case 
of what I have proposed {Principles of Science, p. 28; ist ed., 
vol. i., p. 34) to call a substantial term, but which I find 
that Burgersdyk, Heereboord, and the older logicians called 
a totum homogeneum, the parts being of the same name 
and nature with the whole. (Heereboord, Synopsis Logicae, 
p. 83. See also Mind, vol. i. p. 210.) 

Oily is a noun-adjective, and is concrete, general, positive, 
connotative, as denoting oil and implying the attributes of 
oiliness, doubtfully relative, syncategorematic, 1680. 

Oiliness, noun-substantive, abstract, singular, positive, 
categorematic. 

Where distinctions are omitted, it may be understood that 
they are regarded as inapplicable. 



8 DOCTRINE OF TERMS. [CHAP. i. 

22. Describe the logical characters of the terms 
Related, relative, relation, relativeness, rela 
tionship, relativity. 

I have already dwelt, in the Elementary Lessons (p. 25), 
on the prevalent abuse of the word relation, and other like 
abstract terms. Nothing is more nearly impossible than to 
reform the popular use of language ; but I will point out 
once again that relation is properly the abstract name of the 
connection or bearing of one thing to another, this being 
an attribute of those things. The things in question are 
properly said to be related, or to be relatives. Thus, 
fathers, brothers, sisters, aunts, and cousins, are all relatives 
not relations. Relationship is an abstract term signifying 
the attribute of being related ; it was invented to replace 
relation when this was wrongly used as a concrete term. 
The relationship between a mother and her daughter is 
simply the relation which exists between two such related 
persons or relatives. Relativeness is an uncommon term 
sometimes used to replace the abstract sense of relation, 
where the case is not one of family relation. Relativity is 
a further abstract term, probably due to Coleridge, and of 
which the metaphysicians had better have the monopoly. 



CHAPTER II. 

QUESTIONS AND EXERCISES RELATING TO TERMS. 

i. DESCRIBE the logical characters of the following terms, 
classifying them according as they are 

(a) Abstract or Concrete. 

(b) General or Singular. 

(c) Collective or Distributive. 

(d) Positive or Negative. 

(e) Absolute or Relative. 

(/) Categorematic or Syncategorematic. 

Prime Minister Biped 

Institution Saturn 

Copper Bismarck 

Shameful Monarch 

The London Library Unuseful 

Collection The Times 

School Board Paper 

Deaf Augustus de Morgan 

Equation John Jones 

Innumerous John 

Purpose Triangle 

Function Musicalness 

Cousin Board School 

The Absolute Needlepoint 

Black Representation 



io DOCTRINE OF TERMS. [CHAP. 

Europe Advocate 

Injustice Being 

Brace of partridges Whale 

Dumbness Lawyer 

Planetary System Time 

Classification Manchester 

2. In the case of the following terms distinguish with 
special care between those which are abstract and those 
which are concrete 

Nature Animal Ethericity 

Natural Animalism Scarce 

Naturalness Animality Scarcity 

Naturalism Animalcule Scarceness 

Author Ether Truth 

Authority Ethereal Trueness 

Authorship Etherealness Verity 

3. Investigate the ambiguity of any of the following 
terms as regards their concrete or abstract character 

Weight Science 

Time Schism 

Intention Space 

Vibration Relation 

4. Supply the abstract terms corresponding to the 
following concrete terms 

Wood Conduction 

Stone Atmosphere 

Conduct Alcohol 

Witness Axiom 

Equal Gas 

Table Fire 

Boy Socrates 



ii.] QUESTIONS AND EXERCISES. n 

5. In the case of such of the following terms as you 
consider to be abstract, name the corresponding concrete 
terms 

Analysis Nation 

Psychology Vacuity 

Extension Realm 

Production Folly 

Socialism Evidence 

6. Do abstract terms admit of being put in the plural 
number ? Distinguish between the terms which are abstract 
and concrete in the following list, and at the same time 
indicate which can in your opinion be used in the plural : 
colour, redness, weight, value, quinine, equation, heat, 
warmth, hotness, solitude, whiteness, paper, space. [c.] 

7. Investigate the logical characters and ambiguities of 
the term form in all the following expressions : a religion 
of forms ; the human form ; a form of thought ; a school 
form ; a mere form ; a printer s form ; a form of govern 
ment j form of prayer ; good form ; essential form. 



8. What error is there in the following descriptions ? 

Peerless syncategorematic, general, abstract, positive, 

relative. 
Bacon equivocal, concrete, general, substantial, positive, 

relative. 
Black categorematic, abstract, general, negative, abso- 

1nf/=> 



9. Analyse the following sentences as regards the logical 
character of each term found in them, distinguishing 
especially between such as are concrete or abstract, 
collective or distributive, singular or general 



12 DOCTRINE OF TERMS. [CHAP. 

Logic is the science of the formal laws of thought. 

Entre 1 homme et le monde il faut 1 humanite. 

Art is universal in its influence ; so may it be in its 
practice, if it proceed from a sincere heart and a quick 
observation. In this case it may be the merest sketch, or 
the most elaborate imitative finish. 

10. Burton, in his Etmscan Bologna, p. 234, uses the 
abstract term Etruscanidty. Is it possible in like manner 
to make an abstract term corresponding to every concrete 
one ? If so, supply abstracts for the following concretes- 
Sir Isaac Newton. Royal Engineers. 
Dictionary. Postal Telegraph. 

11. What logical faults do you detect in the following 
expressions ? 

The standard authorship of modern times. 

The three great nationalities of Western Europe. 

The legal heir is not necessarily a man s nearest relation. 

That unprincipled notoriety Pietro Aretino. 

12. Coleridge, in a celebrated note to his Aids to Reflec 
tion, thus defines an Idea : An Idea is the indifference of 
the objectively real and the subjectively real : so, namely, 
that if it be conceived as in the Subject, the idea is an 
Object, and possesses objective truth ; but if in an Object, 
it is then a Subject, and is necessarily thought of as exer 
cising the powers of a Subject. Thus an Idea, conceived 
as subsisting in an Object, becomes a Law : and a law 
contemplated subjectively (in a mind) is an Idea? 

Analyse the meanings of the terms Idea, Object, Subject, 
Real, Truth, Law, etc., in the above passage, with respect 
especially to their concreteness or abstractness. [L.] 



IL] QUESTIONS AND EXERCISES. 13 

13. Name the negative terms which correspond to the 
following positive terms 

Illumination Variable 

White Famous 

Certain Notorious 

Constant Valid 

Dying Plenty 

14. Name the positive terms which correspond to the 
following negative or apparently negative terms 

Immensity Falsehood 

Inestimable Unravelled 

Disestablishment Infamous 

Unpleasant Presuppositionless 

Want Shameless 

Unloosed Empty 

Indifferent Intact 

Headless Ignominious 

15. In examining the following list of terms, distinguish, 
as far as possible, between those which are really negative 
in form and origin, and those which only simulate the 
character of negatives 

Annulled Undespairing 

Disannulled Invalid 

Antidote Headless 

Infrequent Independence 

Eclipse Individual 

Undisproved Indolent 

The Infinite Disagreeable 

Impassioned Despairing 

Immense Infant 

Purposeless Deafness 



i 4 DOCTRINE OF TERMS. [CHAP. 

1 6. Can you find any examples of terms in the dictionary 
which are true double negatives ? Paired, Impaired, 
and Unimpaired, may perhaps be affirmed respectively 
of two things which are equal, unequal, and not unequal. 
Analyse the meaning of each of the following terms, and 
show whether it is or is not a true double negative 

Indefeasible Indefatigable 

Uninvalided Uninjured 

Undecomposable Undecipherable 

Undefaceable Undeformed 

Indestructible Indistinguishable 

17. How are the denotation and connotation of a con 
crete term related to the denotation of the corresponding 
abstract term ? 

1 8. Explain the difference of denotation and connotation 
with reference to the terms Law, Legislator, Legality, 
Crime. [L.] 

19. Compare the connotation of the following sets of 
terms 

J Abbey j Caesar 

( Westminister Abbey ( Roman 

( Mineral c Road 

< Oxide of iron < Means of communication 

( Ore V Railway 

20. Distinguish in the following list such terms as are 
non-connotative, naming at the same time the logician 
whose opinion on the subject you adopt 

Virtue Socrates 

Virtuous Barmouth 

The mother of the Gracchi 

Gladstone The Lord Chamberlain 



ii.] QUESTIONS AND EXERCISES. 15 

21. Form a list of twelve purely non-connotative names. 

22. What is, if any, the connotation of these terms: 
Charles the First ; Richelieu ; John Smith ; Santa Maria 
Maggiore ? 

23. Try to name half-a-dozen perfectly non- relative 
names, and then inquire whether they really are non-relative. 
What is the relation implied or involved in each of the 
following terms ? 

Metropolis County 

Realm Alphabet 

Capital city Sun 

24. Show, by examples, that the division of Names into 
general and singular does not coincide with the division 
into abstract and concrete. [L.] 

25. What kinds of words can stand as the subject of a 
proposition, and what kinds are excluded ? [o.] 

26. Distinguish between the distributive, collective, or 
singular use of these Latin adjectives of quantity : omnis, 
omnes, cunctus, cuncti, ullus, quidam, aliquis. 

27. What is peculiar about the use of certain terms in 
the following extracts ? 

(1) Frenchmen, I ll be a Salisbury to you. 

(2) His family pride was beyond that of a Talbot or a 

Howard. 

(3) In quo quisque artificio excelleret, is in suo genere 

Roscius diceretur. 

(4) When foe meets foe. 

28. How does Logic deal with verbs, adverbs, and con 
junctions ? 

29. How many logical terms are there in the following 
witty epigram ? Which and what are they ? 



16 DOCTRINE OF TERMS. [CHAP. 

What is mind ? No matter. 
What is matter ? Never mind. 

30. How many logical terms are there in each of the 
following sentences? Ascertain exactly how many words 
are employed in each such term. 

(1) The Royal Albert Hall Choral Society s Concert is 
held in the Albert Hall on the Kensington Gore Estate 
purchased by the Royal Commissioners of the Great 
Exhibition of 1851. 

(2) A name is a word taken at pleasure to serve for a 
mark which may raise in our mind a thought like to 
some thought we had before, and which being pro 
nounced to others, may be to them a sign of what 
thought the speaker had before in his mind. 

31. Words, says Hobbes, are insignificant (that is without 
meaning), when men make a name of two names, whose 
significations are contradictory and inconsistent : as this 
name, an incorporeal body. 

The following are a few instances of such apparently self- 
inconsistent names, and the student is requested to add to 
the list 

(1) Corporation sole. 

(2) Trigeminus. 

(3) Manslaughter of a woman. 

(4) An invalid contract. 

(5) A breach of a necessary law of thought. 

32. How would you explain the following apparent 
absurdities ? 

An Act of Parliament (1798-99) prohibited the importa 
tion of * French lawns not made in Ireland. 



ii.] QUESTIONS AND EXERCISES. 17 

Ferguson (History of Architecture, Vol. II., p. 233) de 
scribes a certain Moabite tower as a square Irish round 
tower. 

33. Are the following terms perfectly univocal or un 
ambiguous, or can you point out any equivocation which is 
possible in their use ? 

Penny Lecture-Room 

Charcoal Victoria Street 

Aluminium Bible 

Second Monday 

34. Trace out and explain the ambiguities which affect 
any of the following terms 

Organ Stone March 

Sole Corn Mood 

Ear Diet Mean 

Bowl Perch Force 

Rock (stone) Bole Bowl 

Rock (bird) Strait Straight 

35. Draw out complete lists of all the words or expres 
sions which have been developed out of the roots of the 
following words (see Elementary Lessons in Logic, pp. 
32 36, and Lesson VI.) 

Post Logic 

Section Faction 

Final Function 

Mission Decline 



CHAPTER III. 



KINDS OF PROPOSITIONS. 



i. IN this chapter propositions will be described and 
classed according to the ancient Aristotelian doctrine, in 
which four principal forms of propositions were recognised, 
thus tabularly stated : 



Affirmative. 



Negative. 



Universal 



Particular 



Symbol = A 
All X is Y 


Symbol = E 
No X is Y 


Symbol = I 
Some X is Y 


Symbol = O 
Some X is not Y 



Singular propositions are to be classed as universal, and 
indefinite propositions, in which no indication of quantity 
occurs, must be interpreted at discretion as universal or 
particular. The student is supposed to be familiar with 
what the ordinary text-books say upon the subject. 

I first give a series of Examples of propositions, with 
brief comments upon their logical form and peculiarities. 
A copious selection of exercises is then supplied in the 
next chapter for the student to treat in like manner. 



CHAP, in.] KINDS OF PROPOSITIONS. 19 

EXAMPLES. 

2. * Books are not absolutely dead things. O. 

This proposition is indefinite or pre-indesignate, as 
Hamilton would call it (Lectures on Logic, Vol. I. (III.), 
p. 244); but, as we can hardly suppose Milton to have 
thought that all books were living things, I take it to 
mean some books are not, etc., that is to say, particular 
negative. 

3. The weather is cold. A. 

The weather means the present state of the surrounding 
atmosphere, and may be best described as a singular term, 
which makes the assertion universal. 

4. Not all the gallant efforts of the officers and escort 
of the British Embassy at Cabul were able to save 
them. E. 

At first sight this seems to be a particular negative, like 
Not all that glitters is gold ; but a little consideration 
shows that gallant efforts is a collective whole, the efforts 
being made in common, and therefore either successful or 
unsuccessful as a whole. The meaning then is, The whole 
of the gallant efforts, etc., were not able to saVe the men. 
It is a universal negative. 

5. * One bad general is better than two good ones. A. 

This saying of Napoleon looks at first like a particular or 
even a singular proposition ; but the one bad general 
means not any definite one, but any one bad general 
acting alone. 

6. No non-metallic substance is now employed to make 
money. E. 



20 KINDS OF PROPOSITIONS. [CHAP. 

The subject is a negative term, and the proposition might 
be stated as All non-metallic substances are not any of 
those employed to make money. 

7. Multiplication is vexation. 

If all multiplication is so, this is A ; there are certainly 
other causes of vexation. 

8. Wealth is not the highest good. E. 
Affirmatively, wealth is one of the things which are not 

the highest good. 

9. Murder will out. A. 

Like most proverbs, this is an unqualified universal 
proposition ; its material truth may be doubted. 

10. A little knowledge is a dangerous thing. A. 
This looks like a particular affirmative, but is really 

A, as meaning that any small collection of knowledge 
is, etc. 

11. All these claims upon my time overpower me. A. 
Dr. Thomson points out (Outline, 5th ed., p. 131) that 

all is here clearly collective. 

12. The whole is greater than any of its parts. A. 
Though apparently singular, this is really a general axiom, 

meaning any whole is greater, etc. 

13. No wolves run wild in Great Britain at the present 
day. E. 

1 4. Who seeks and will not take, when once tis offered, 
shall never find it more. E. 

This seems to be a compound proposition, but the sub 
ject is, Any one who is seeking, but has not taken when 
once it was offered. 



in.] EXAMPLES. 21 

15. The known planets are now more than a hundred 
in number. A. 

Clearly a collective singular affirmative proposition, and 
therefore universal. Of course the planets separately could 
not have the predicate here affirmed. 

1 6. Figs come from Turkey. I. 

Indesignate; that is to say, we cannot assume without 
express statement that it is intended to say, All figs come 
from Turkey. 

17. Xanthippe was the wife of Socrates. A. 

1 8. No one is free who is enslaved by his appe 
tites. E. 

19. Certain Greek philosophers were the founders of 
logic. A. 

Apparently I ; but if certain means a certain definite 
group of men, each of whom was essential in his time, 
the proposition becomes collective and singular, hence 
universal. 

20. Comets are subject to the law of gravitation. A. 

Indefinite affirmative ; but in a matter of such universality 
it may be interpreted as A. 

21. Democracy ends in despotism. I. 

Again indefinite ; but as referring to matter in which no 
rigorous laws have been detected it should be interpreted 
particularly. 

22. Men at every period since the time of Aristotle have 
studied logic. I. 

Obviously particular as regards men. 



22 KINDS OF PROPOSITIONS. [CHAP. 

23. Few men know how little they know. O. 
That is, Most men do not know, etc. Hence O. 

24. Natura omnia dedit omnibus. A. 

Singular affirmative, because natura is a singular term. 
The assertion is one of Hobbes , and is thoroughly am 
biguous as regards omnia and omnibus, which might be 
capable either of collective or distributive meaning. No 
doubt, however, the meaning is that Nature did not assign 
anything to any particular person ; if so, both must be taken 
collectively. 

25. There are many cotton-spinners unemployed. I. 

Really a kind of numerical assertion ; but if to be classed 
at all, it must be I, many being only a part of all. 

26. A^few Macedonians vanquished the vast army of 
Darius. A. 

Collective singular affirmative, because the few of course 
acted together. It is a question whether the predicate is 
not also singular. 

27. True Faith and Reason are the soul s two eyes. A. 
Collective singular. 

28. A perfect man ought always to be busy conquering 
himself. A. 

All perfect men ought, etc. 

29. A truly educated man knows something of every 
thing and everything of something. A. 

There seems to be two predicates, and hence a compound 
sentence; but this is not the case, because the truly educated 
man must know both. 



in.] EXAMPLES. 23 

30. Some comets revolve in hyperbolic orbits. I. 
Particular affirmative as it stands. 

31. The dividends are paid half-yearly. A. 
* The dividends includes all so known. 

32. Ov TO peya ev ecrrt, TO 8 ev /xeya. O and A. 

This must mean that not all great things are good (O), 
but that all good things are great (A). There are three 
classes of things great and good ; great and not-good ; 
not-great and not-good. 

33. It is force alone which can produce a change of 
motion. A. 

It = what can produce, etc. The meaning is, Whatever 
produces a change of motion is some kind of force ; but 
there is no assertion that force = whatever produces, etc. 

34. We have no king but Caesar. 

As it stands, A ; but the meaning conveyed implies that 
Caesar is our king ; Nobody who is not Caesar is our 
king. 

35. It is true that what is settled by custom though it 
be not good, yet at least it is fit. 

Complex; three propositions in all. 

36. God did not make man, and leave it to Aristotle to 
make him rational. 

A simple and a singular negative proposition ; the not 
applies to all that follows conjunctively, for of course Locke 
could not have intended to assert that God did not make 
man. * E. 



24 KINDS OF PROPOSITIONS. [CHAP. in. 

37. * Dublin is the only city in Europe, save Rome, which 
has two cathedrals. 

Compound sentence implying three propositions, namely 
Dublin has two cathedrals. A. 
Rome has two cathedrals. A. 
All European cities, not being Dublin and not being 
Rome, have not two cathedrals. E. 

38. The affections are love, hatred, joy, sorrow, hope, 
fear, and anger. 

Really a disjunctive proposition. Affection is either love, 
or hatred, or, etc. This implies that love is an affection, 
hatred is an affection, etc. 



CHAPTER IV. 

EXERCISES IN THE DISCRIMINATION OF PROPOSITIONS. 

i. EXAMINE each of the following propositions, and point 
out in succession 

(a) Which is the subject. 

(b) Which is the predicate. 

(c) Whether the proposition is affirmative or negative. 

(d) Whether it appears to be universal or particular. 

(e) Whether there is ambiguity or other peculiarity 

in the proposition. 

(1) All foraminifera are marine organisms. 

(2) They never pardon who have done the wrong. 

(3) Great is Diana of the Ephesians. 

(4) No mammalia are parasites. 

(5) Non progredi est regredi. 

(6) Not every one can integrate a differential equation. 

(7) All, all are gone, the old familiar faces. 

(8) He that is not for us is against us. 

(9) "Apurrov /xev vStop. 

(10) Men mostly hate those whom they have injured, 

(n) Old age necessarily brings decay. 

(12) Nothing morally wrong is politically right. 

(13) What I have written I have written. 

(14) It is not good for man to be alone. 



26 PROPOSITIONS. [CHAP. 

(15) A certain man had a fig-tree, 
(l 6) XaAeTra rot KaAa. 

(17) There s something rotten in the state of Denmark. 

(18) To be or not to be, that is the question. 

(19) Ye are my disciples, if ye do all I have said unto you. 

(20) Possunt qui posse videntur. 

(21) There can be no effect without a cause. 

(22) Rien n est beau que le vrai. 

(23) Pauci laeta arva ten emus. 

(24) All cannot receive this saying. 

(25) Fain would I climb, but that I fear to fall. 

(26) There s not a joy the world can give like that it 

takes away. 

(27) Not to know me argues thyself unknown. 

(28) Two blacks won t make a white. 

(29) Few men are free from vanity. 

(30) He that fights and runs away may live to fight 

another day. 

(31) We are what we are. 

(32) There is none good but one. 

(33) Two straight lines cannot inclose space. 

(34) Better late than never.. 

(35) Cruel laws increase crime. 

(36) Omnes omnia bona dicere. 

(37) Le genie n est qu une plus grande aptitude a la 

patience. 

(38) Whosoever is delighted in solitude is either a wild 

beast or a god. 

(39) Summum jus summa injuria. 

(40) Non omnes moriemur inulti. 

(41) Haud ignara mali miseris succurrere disco. 

(42) Familiarity breeds contempt. 

(43) Some politicians cannot read the signs of the times. 



iv.J EXERCISES. 27 

(44) Only the ignorant affect to despise knowledge. 

(45) Recte ponitur; vere scire esse per causas scire. 

(46) Only Captain Webb is able to swim across the 

Channel. 

(47) Some books are to be read only in parts. 

(48) E pur si muove. 

(49) Civilisation and Christianity are co-extensive. 

(50) Some men are not incapable of telling falsehoods. 

(51) Sunt nonnulli acuendis puerorum ingeniis non 

inutiles lusus. 

(52) All is not true that seems so. 

(53) Me miserable. 

(54) The Claimant, Arthur Orton, and Castro are in all 

probability the same person. 

(55) The three angles of a triangle are necessarily equal 

to two right angles. 

(56) Many rules of grammer overload the memory. 

(57) Nullius exitium patitur natura videri. 

(58) Summse artis est occultare artem. 

(59) Wonderful are the results of science and industry 

in recent years. 

(60) Love is not love which alters when it alteration finds. 

(6 1) A healthy nature may or may not be great ; but there 

is no great nature that is not healthy. 

(62) Quas dederis solas semper habebis opes. 

(63) Quod volunt, id credunt homines. 

(64) IIa(ra orap ov 8tKGua>$^o-eTai. 

(65) Antiquitas seculi, juventus mundi. 

(66) That would hang us, every mother s son. 

(67) Men in great place are thrice servants. 

(68) Justice is ever equal. 

(69) A friend should bear a friend s infirmities. 

(70) Men are not what they were. 



28 PROPOSITIONS. [CHAP. 

(71) The troops took one hour in passing the saluting 

point. 

(72) Nemo mortalium omnibus horis sapit. 

(73) Fugaces labuntur anni. 

(74) ATJTOS eyw et/jit. 

(75) Communia sunt amicorum inter se omnia. 

(76) Dictum sapienti sat est. 

(77) The Romans conquered the Carthaginians. 

(78) The fear of the Lord, that is wisdom. 

(79) To live in hearts we leave below is not to die. 

(80) Tis only noble to be good. 

(81) Dum spiro spero. 

2. In looking over the following list of propositions 
distinguish between those which have a distributive and 
those which have a collective subject. 

(1) All the asteroids have been discovered during the 

present century. 

(2) All Albinos are pink-eyed people. 

(3) The facts of aboriginal life seem to indicate that 

dress is developed out of decorations. 

(4) Non omnes omnia decent. 

(5) Dirt and overcrowding are among the principal 

causes of disease. 

(6) Omnes apostoli sunt duodecim. 

(7) Many artisans are unemployed. 

(8) The side and diagonal of a square are incommen 

surable. 

(9) Omnis homo est animal. 

(10) Nihil est ab omni parte beatum. 

3. Ascertain exactly how many distinct assertions are 
made in each of these sentences, and assign the logical 
characters of the propositions. 



iv.] EXERCISES. 29 

(1) Tis not my profit that doth lead mine honour : mine 

honour, it. 

(2) True, tis a pity; pity tis, tis true. 

(3) Hearts, tongues, figures, scribes, bards, poets, cannot 

think, speak, cast, write, sing, number, ho ! his 
love to Antony. 

(4) A horse, a horse ! my kingdom for a horse. 

(5) Istuc est sapere, non quod ante pedes modo est 

videre : sed etiam ilia, quae futura sunt, prospicere. 

(6) Virtue consists neither in excess nor defect of action, 

but in a certain mean degree. 

(7) The glories of our blood and state are shadows, not 

substantial things. 

(8) To gild refined gold, to paint the lily, 

To throw a perfume on the violet, 

To smooth the ice, or add another hue 

Unto the rainbow, or with taper light 

To seek the beauteous eye of heaven to garnish 

Is wasteful and ridiculous excess. 

(9) All places that the eye of heaven visits, 
Are to a wise man ports and happy havens. 

(TO) The age of chivalry is gone, and the glory of 

Europe extinguished for ever, 
(n) Poeta nascitur, non fit. 

(12) Not all speech is enunciative, but only that in which 

there is truth or falsehood. 

(13) Devouring Famine, Plague, and War, 

Each able to undo Mankind, 
Death s servile emissaries are. 

(14) Many are perfect in men s humours, that are not 

greatly capable of the real part of business, which 
is the constitution of one that hath studied men 
more than books. 



3 o PROPOSITIONS. [CHAP. iv. 

(15) Vivre, ce n est pas respirer, c est agir. 

(16) Justice is expediency, but it is expediency speaking 

by general maxims, into which reason has concen 
trated the experience of mankind. 

(17) Men, wives, and children, stare, cry out, and run as 

it were doomsday. 

4. Distinguish so far as you can between the propositions 
in the following list which are to you explicative and ampli- 
ative. (See Elementary Lessons, pp. 68 69. Thomson s 
Outline of the Necessary Laws of Thought, 81.) 

(1) Homer wrote the Iliad and Odyssey. 

(2) A parallelepiped is a solid figure having six faces, of 

which every opposite two are parallel. 

(3) The square on the hypothenuse of a right-angled 

triangle is equal to the sum of the squares on the 
sides containing the right angle. 

(4) The swallow is a migratory bird. 

(5) Axioms are self-evident truths. 

5. Classify the following signs of logical quantity accord 
ing as they are generally used to indicate universality, 
affirmative or negative, or particularity, affirmative or 
negative 

Several, none, certain, few, ullus, nullus, nonnullus, not a 
few, many, the whole, almost all, not all. 



CHAPTER V. 

CONVERSION OF PROPOSITIONS, AND IMMEDIATE INFERENCE. 

i. THE student is referred to the Elementary Lessons in 
Logic, or to other elementary text-books, for the common 
rules of conversion and immediate inference, but for the 
sake of easy reference, the ancient square of opposition is 
given below. 

A Contraries .... .E 



C> 









i Subcontraries . :o 

All the relations of propositions and the methods of 
inference applying to a single proposition will be found fully 
exemplified and described in the following questions and 
answers. 



32 CONVERSION. [CHAP. 

2. It appears to be indispensable, however, to endeavour 
to introduce some fixed nomenclature for the relations of 
propositions involving two terms. Professor Alexander 
Bain has already made an innovation by using the name 
obverse, and Professor Hirst, Professor Henrici and other 
reformers of the teaching of geometry have begun to 
use the terms converse and obverse in meanings incon 
sistent with those attached to them in logical science 
(Mind, 1876, p. 147). It seems needful, therefore, to state 
in the most explicit way the nomenclature here proposed 
to be adopted with the concurrence of Professor Robertson. 

Taking as the original proposition all A are BJ the 
following are what we may call the related propositions 

INFERRIBLE. 

Converse. Some B are A. 
Obverse. No A are not B. 
Contrapositive. No not B are A, 
or, all not B are not A. 

NON-INFERRIBLE. 

Inverse. All B are A. 
Reciprocal. All not A are not B. 

It must be observed that the converse, obverse, and con- 
trapositive are all true if the original proposition is true. 
The same is not necessarily the case with the inverse and 
reciprocal. These latter two names are adopted from the 
excellent work of Delbceuf, Prolegomenes Philosophiques de 
la Geometric, pp. 8891, at the suggestion of Professor 
Groom Robertson. (Mind, 1876, p. 425.) 



v.] QUESTIONS AND ANSWERS. 33 



QUESTIONS AND ANSWERS. 

3. Give all the logical opposites of the proposition, 
All metals are conductors. 

This is a universal affirmative proposition, having the 
symbol A. By its logical opposites we mean the corre 
sponding propositions in the forms E, I, and O, which 
have the same subject and predicate, and are related to it 
respectively as its contrary, contradictory, and subaltern, in 
the way shown in the Logical Square (p. 31) and explained 
in many Manuals. These opposite propositions may be 
thus stated 

Subaltern (I) Some metals are conductors. 

Contradictory (O) Some metals are not conductors. 
Contrary (E) No metals are conductors. 

The first of these (I) may be inferred from the original ; 
the other two (O and E), so far from being inferrible, are 
inconsistent with its truth. 

4. Given that a particular negative proposition 
is true, is the following chain of inferences 
correct ? O is true, A is false, I is false, and 
therefore E is true. If so, the truth of O 
involves the truth of E. 

There is a false step in this argument ; for the falsity of 
A does not involve the falsity of I. It may be (and is 
materially false) that all men are dishonest ; but it never- 



34 CONVERSION. [CHAP. 

theless may remain true that some men are dishonest 
Observe, then, that the falsity of A does not involve the 
truth of I, nor does the truth of I involve the truth or 
falsity of A. But the truth of A necessitates that of I. 
As stated in the Elementary Lessons (p. 78), Of sub 
alterns, the particular is true if the universal be true : 
but the universal may or may not be true when the 
particular is true. 

5. How do you convert universal affirmative 
propositions ? 

They must be converted by limitation or per accidens^ 
as it is called, that is to say, while preserving the affirmative 
quality, the quantity of the proposition must be limited from 
universal to particular^ Thus A is converted into I, as 
in the following more or less troublesome instances, the 
Convertend standing first and the Converse second in each 
pair of propositions : 

f All organic substances contain carbon. 

1 Some substances containing carbon are organic. 

f Time for no man bides. 

( Something biding for no man is time. 

j The poor have few friends. 

1 Some who have few friends are poor. 

C A wise man maketh more opportunities than he finds. 
< Some who make more opportunities than they find are 
v wise men. 

C They are ill discoverers who think there is no land, when 
< they can see nothing but water. 
\ Some ill discoverers think there is no land, etc. 



v.] QUESTIONS AND ANSWERS. 35 

( Great is Diana of the Ephesians. 

I Some great being is Diana of the Ephesians. 

( Warm-blooded animals are without exception air-breathers. 

"\ Air-breathers are (with or without exception) warm- 

v blooded animals. 

6. How would you convert Brutus killed 
Caesar? 

The strictly logical converse is Some one who killed 
Caesar was Brutus. For, though a man can only be killed 
once, and Brutus is distinctly said to be the killer, yet in 
formal logic we know nothing of the matter, and Caesar 
might have been killed on other occasions by other persons. 
An absurd illustration is purposely chosen in the hope that 
it may assist to fix in the memory the all-important truth 
that in logic we deal not with the matter. 

7. How do you convert particular affirmative 
propositions ? 

To this kind of proposition simple conversion can be 
applied ; that is to say, the converse will preserve both the 
quantity and the quality of the convertend. In other 
words, I when converted gives another proposition in I ; 
thus either of the following pairs is the simple converse of 
the other : 

(" Some dogs are ferocious animals. 

(. Some ferocious animals are dogs. 

f Some men have not courage to appear as good as they are. 

s Some, who have not courage to appear as good as they 

v are, are men. 

f Some animals are amphibious. 

\ Some amphibious beings are animals. 



36 CONVERSION. [CHAP. 

8. How do you convert universal negative pro 
positions ? 

These also are converted simply, giving another universal 
negative proposition. E gives E. The reason is that both 
the terms of E are distributed ; a universal negative asserts 
complete separation between the whole of the subject and 
the whole of the predicate. No man is a tailed animal 
asserts that not any one man is found anywhere in the class 
of tailed animals. Hence it follows evidently that no one 
being belonging to the class of tailed animals is found in 
the class of men, which result we assert in the simple con 
verse proposition, no tailed animal is a man. Further 
examples of the same mode of conversion are given below. 

f No virtue is ultimately injurious. 

I No ultimately injurious thing is a virtue. 

C No wise man runs into heedless danger. 

\ No one who runs into heedless danger is a wise man. 

/ People will not look forward to posterity who never look 
J backward to their ancestors. 

I People never look backward to their ancestors who will 
v not look forward to posterity. 

Whatever is insentient is not an animal. 
Whatever is an animal is not insentient. 

9. How do you convert particular negative 
propositions ? 

Difficulty arises about this question, because the first 
rule of conversion tells us to preserve the quality of the 
proposition ; the converse accordingly should be negative. 
But a negative proposition always distributes its predicate, 



v.] QUESTIONS AND ANSWERS. 37 

because a thing excluded from a class must be excluded 
from every part of the class. Now the subject of O being 
particular and indefinite, it cannot stand as a distributed 
predicate. It is still possible to say with material truth, 
some men are not soldiers ; but converted this gives the 
absurd result, all soldiers are not men ; or, no soldiers 
are men. Even if we insert the mark of quantity some 
before the predicate, and say, all soldiers are not some 
men, we must remember that some is perfectly indefinite, 
and may include all. The question will be more fully dis 
cussed further on, but, so far as I can see, the particular 
negative proposition, so long as it remains negative and 
indefinite in meaning, is incapable of conversion. This fact 
constitutes a blot in the ancient logic. 

Nevertheless the proposition O is capable of giving a 
converse result when we change it into the equivalent affirma 
tive proposition. If some men are excluded from the 
class * soldiers/ they are necessarily included in the class 
non-soldiers, or, some men are non-soldiers. This is a 
proposition in I, and by simple conversion, as already de 
scribed, gives a converse also in I, some non-soldiers 
are men. As further examples take 

( Some dicotyledons have not reticulate leaves. 

1 Some plants with non-reticulate leaves are dicotyledons. 

J Some crystals are not symmetrical. 

( Some unsymmetrical things are crystals. 

J All men have not faith. 

( Some who have not faith are men. 

/ Not every one that saith unto me, Lord, Lord, shall enter 

\ into the Kingdom of Heaven. 

^ Some who shall not enter into the Kingdom of Heaven 

\ say unto me, Lord, Lord. 



38 CONVERSION. [CHAP. 

10. How do you convert singular propositions ? 

Singular propositions, being those which have a singular 
term as subject, may be divided into two classes, according 
as the predicate is a singular or a general term. (See 
Karslake, 1851, vol. i., p. 54.) The former will always be 
converted simply, one single thing being identified with the 
same under another name, as in Queen Victoria is the 
Duchess of Lancaster, converted into the Duchess of 
Lancaster is Queen Victoria. Simple conversion will also 
apply if the predicate be a general term, provided that the 
proposition be negative so as to distribute this term. Thus, 
St. Albans is not a great city becomes * no great city is 
St. Albans. But if the predicate be general and undis 
tributed, as in an affirmative singular proposition, then we 
must convert per accidens, and limit the new subject to some 
or even one significate of the general term. Examples of 
each case follow : 

( The better part of valour is discretion. 
( Discretion is the better part of valour. 

J Time is the greatest innovator. 
1 The greatest innovator is time. 

f London is the greatest of all cities. 
(. The greatest of all cities is London. 

{London is not a beautiful city. 
No beautiful city is London. 

( Le style est 1 homme meme. 
1 L homme meme est le style. 

( All the allied troops fought courageously. 

1 Some who fought courageously were the allied troops. 



v.] QUESTIONS AND ANSWERS. 39 

( Mercy but murders, pardoning those that kill. 
{ Something which murders is mercy, pardoning those that 
V kill. 

(Not all the figures that Babbage s calculating machine 
could run up, would stand against the general heart. 
Something which would not stand against the general heart 
is all the figures (collectively) that Babbage s machine 
could run up. 

II. Show how to convert the propositions 

(1) All mathematical works are not difficult 

(2) All equilateral triangles are equiangular. 

(3) No triangle has one side equal to the 

other two. 

The first proposition, as it stands, is ambiguous, for it 
looks like the universal negative, no mathematical works 
are difficult. But, according to custom, we may interpret 
it as meaning that not all mathematical works are difficult, 
or some mathematical works are not difficult, a proposition 
in the form O. This cannot be converted simply, as already 
explained (p. 36), because we must preserve the negative 
quality, and all (or some) difficult things are not mathe 
matical works being negative would distribute its predicate 
mathematical works. We can, however, make O into I, 
some mathematical works are not-difficult things, and we 
can convert this simply into some not-difficult things are 
mathematical works. 

Proposition (2), as it stands, is in A, and can only be 
logically converted by limitation into some equiangular 
triangles are equilateral. Geometrically it could easily be 
shown that the inverse proposition all equiangular triangles 
are equilateral, is also true ; but we must of course not 



40 CONVERSION. [CHAP. 

allow knowledge of the matter in question to influence us 
in logical deduction, and the inverse proposition cannot 
be inferred from the original. 

Number (3) is a universal negative, and must be converted 
simply into * Nothing having one side equal to the other two 
is a triangle; but there is something paradoxical about 
this result, which the student is recommended to investigate. 

12. Convert Life every man holds dear. 

This is an example given in the Elementary Lessons, 
(p. 304). Students have variously converted it into 

Life is held dear by every man. 

Some life is held dear, etc. 

No man holds death dear (!) 

and so forth. But it ought surely to be easy to see that the 
grammatical object is transposed, life being the object of 
holds dear. The statement is that every man holds life 
dear, and is explicitly a universal affirmative proposition, 
to be converted by limitation into some who hold life dear 
are men. 

13. Convert the proposition It rains. 

What is it that rains ? What is it ? Surely the environ 
ment, or more exactly the atmosphere. The proposition 
then means the atmosphere is letting rain fall. The con 
verse will therefore be something which is letting rain 
fall is the atmosphere. But in this and many other cases 
the Aristotelian process of conversion by limitation gives a 
meaningless if not absurd result. 

14. Convert the proposition He jests at scars 
who never felt a wound. 

This is the 8th example on p. 304 of the Elementary 



v.] QUESTIONS AND ANSWERS. 41 

Lessons, and has elicited from time to time some amazing 
efforts at conversion, such as 

Some jests at scars are made by one who never felt 
a wound. 

Scars are jested at by him who, etc. 

Some scars jest at him who never felt a wound, (sic.) 

Some scars are jests to one who, etc. 

The subject of the proposition is of course he who never 
felt a wound, and the proposition asserts that he thus 
described jests at scars. As there is no limitation of 
quantity we may take the subject as universal ; and, although 
there is negation within the subject, the copula is affirmative, 
and the proposition is in the form A. It is thus converted 
by limitation into some who jest at scars are persons who 
have never felt a wound. 

15. Convert the proposition P struck Q. 

To this simple question I have got answers that, since P 
is distributed, and Q undistributed, we must convert by 
limitation, getting some Q struck P ; or by contraposition 
some not- Q struck not-/ . 7 Such blunders and nonsense 
arise from failing to notice that struck is not a simple 
logical copula. There is, of course, a relation between P 
and <2; but as regards P, the proposition simply asserts 
that P is a person who struck Q, possibly not the only 
one. Hence the converse by limitation is some person 
who struck Q is P. 

Not a few examinees would at once convert P struck 
Q into <2 struck /*, but this, although very likely to 
happen materially, is not logically necessary. 

16. What is the obverse of the proposition All 
metals are elements ? 



42 IMMEDIATE INFERENCE. [CHAP. 

The obverse is a new term introduced by Professor 
Alexander Bain, and its meaning is thus described by him 
in his Deductive Logic, pp. 109, no. In affirming one 
thing, we must be prepared to deny the opposite : " the 
road is level," " it is not inclined," are not two facts, but 
the same fact from its other side. This process is named 
OBVERSION. He proceeds to point out that each of the 
four propositional forms, A, I, E, O, admits of an obverse. 
Every X is Y becomes no X is not- K Some X is V 
becomes some X is not not- K No X is Y becomes all 
X is not-K Some X is not Y 1 becomes some X is not- 
y. Accordingly the obverse of the proposition above will 
be No metals are not elements. 

Professor Bain goes on to describe what he calls Material 
Obversion, justified only on an examination of the matter 
of the proposition. Thus from warmth is agreeable, he 
infers, after examination of the subject-matter, that cold is 
disagreeable. If knowledge is good, ignorance is bad. I 
feel sure, however, that this mixing up of so-called material 
obversion with formal obversion is likely to confuse people 
altogether. Indeed, Mr. Bain is himself confused, for he 
cites, I don t like a curving road, because I like a straight 
one as a childish reason, being no reason at all, but the 
same fact in obverse. Now, if there is any relation at all 
between these two propositions, it is certainly a case of 
material obversion ; but in reality they do not express the 
same fact at all. The formal obverse of I like a straight 
road, is I am not one who does not like a straight road. 
We might perhaps infer, I do not dislike a straight road ; 
but there is clearly no reference to curved roads at all. 

While accepting the new term obversion in the sense of 
formal obversion, I must add that students have begun to 
use it with the utmost laxity, confusing the obverse with the 



v.] QUESTIONS AND ANSWERS. 43 

converse, the contrapositive, etc. To prevent logical nomen 
clature from falling into complete chaos, it seems to be 
indispensable to choose convenient names for the simpler 
relations of prepositional forms, as attempted above (p. 32), 
and to adhere to them inflexibly. 

17. What is conversion by contraposition ? Give 
the contrapositive of All birds are bipeds. 

There is nothing which I have found so difficult in 
teaching logic as to get the student to comprehend and 
remember this process of contraposition ; particular attention 
is therefore requested to the above question. 

Having a proposition in A, we get its contrapositive by 
taking the negative of its predicate, and affirming of this as 
a subject the negative of the original subject. Thus, if all 
Xs are Fs, we take all not-Fs as a new subject, and affirm 
of them that they are all not-Xs, getting the proposition all 
not- Fs are not-JTs, which is either A or E, according as 
we do or do not join the negative particle to the predicate X. 
Accordingly the contrapositive of the proposition all birds 
are bipeds will be all that are not bipeds are not birds. 

It is one thing to obtain the contrapositive, another thing 
to see that it may be inferred from the premise. The late 
Professor De Morgan used to hold that the act of inference 
is a self-evident one, and needs no analysis ; but the process 
may certainly be analysed. Thus we may obvert the pre 
mise All Xs are Fs, obtaining No- J^s are not Fs, which 
is a proposition in E, and then convert simply into * No 
not-Fs are Xs, also in E, or else All not-Fs are not Xs. 
The contrapositive, then, is the converse of the obverse. 

We may also prove the truth of the contrapositive 
indirectly; for what is not- F must be either X or not-^Y"; 



44 IMMEDIATE INFERENCE. [CHAP. 

but if it be X it is by the premise also K, so that the same 
thing would be at the same time not- Y and also J 7 , which 
is impossible. It follows that we must affirm of not-K 
the other alternative, not-X. (See Chapter XXI. below; 
also Principles of Science, pp. 83, 84 ; first ed., vol. i., pp. 
97, 98.) 

18. Give the converse of the contrapositive of 
the proposition All vegetable substances are 
organic. 

As learnt from the last question, the contrapositive is 
All not-organic substances are not vegetable substances. 
We may take this to be equivalent to No inorganic sub 
stances are vegetable substances (E), the simple converse 
of which is No vegetable substances are inorganic sub 
stances, the obverse of the premise. But, if we treat the 
contrapositive as a universal affirmative proposition, thus, 
All inorganic substances are non-vegetable substances/ 
we must convert by limitation, getting Some non-vegetable 
substances are inorganic, which is the subaltern of the 
obverse, and cannot by any process of inference lead us 
back to the original. Conversion by limitation is easily 
seen to be a faulty process which always occasions a loss 
of logical force. 

As we shall afterwards observe, this kind of conversion 
introduces a new term, namely the indeterminate adjective 
* some, so that the inference is not really confined to the 
terms of the original premise. Although we may not be 
able to dispense entirely with the word, owing to its employ 
ment in ordinary discourse, we shall ultimately eliminate it 
from pure formal logic, and relegate it to the branch of 
numerical logic. 



v.] QUESTIONS AND ANSWERS. 45 

19. Take the following proposition, all water 
contains air ; convert it by contraposition : 
change the result into an affirmative pro 
position, and convert. 

To show the need of more careful logical training than 
has hitherto been common, even in the great Universities, 
I give a few specimens of answers which I received to the 
above question. The contrapositive of the proposition was 
variously stated, as 

All air does not contain all water. 

All air is not contained in water. 

All not-air is not a thing contained by not-water. 

Some air is not contained in water. 

Some not-air contains no water. 

All not-air contains water. 

The logicians who drew these inferences then proceeded 
by simple conversion to get such results as the following : 

Some water is not without some air. 
No water contains not some air. 
No water contains no air. 

One too clever student inferred that All or every vacuum 
is a void of water, which he converted, simply indeed, into 

Every void of water is a vacuum ! 

An examiner in logic is sometimes forced to believe that 
there is a void in the brains of an examinee ; but the 
absence of any sufficient training in logical work is more 
often the cause of the lamentable results shown above. In 
any case it seems impossible to agree with De Morgan that 
contraposition is a self-evident process. 



46 IMMEDIATE INFERENCE. [CHAP. 

These absurd answers are mainly due to the failure to 
observe that in the proposition All water contains air, the 
two words contains air, form the grammatical predicate, 
comprehending both the logical predicate and the logical 
copula. Logically then the proposition is All water is 
containing air, or All water is what contains air. The 
contrapositive then is All that does not contain air is not 
water. Uniting the negative particle to the predicate 
* water, and converting by limitation, we obtain Some 
not-water is what does not contain air. 

20. Describe the logical relations, if any, between 
each of the following propositions and each 
other 

(1) All organic substances contain carbon. 

(2) There are no inorganic substances which do 

not contain carbon. 

(3) Some inorganic substances do not contain 

carbon. 

(4) Some substances not containing carbon are 

organic. 

Of these, (i) is a universal affirmative, the contrapositive 
of which is All substances not containing carbon are inor 
ganic substances. Hence the converse by limitation of this 
contrapositive is, Some inorganic substances are substances 
not containing carbon, equivalent to (3). 

Proposition (2) is the obverse of All inorganic substances 
contain carbon, which is the contradictory of (3). 

To obtain (4) we must take the contrary of (i), that is, 
no organic substances contain carbon, express it in the 
affirmative form, All organic substances are substances 
not containing carbon, and then convert it by limitation, 



v.] QUESTIONS AND ANSWERS. 47 

21. Take any proposition suitable for the pur 
pose, convert it by contraposition, convert it 
again simpliciter, change the result into an 
affirmative proposition, and show that you 
may regain the original proposition. [c.] 

The most suitable kind of proposition for the purpose 
will be a universal affirmative, such as 

(1) All birds are bipeds. 

The contrapositive may be stated in the form of E. 

(2) No not-bipeds are birds. 

Which is converted simpliciter into E, the obverse of (i). 

(3) No birds are not-bipeds. 

When thrown into the affirmative form by a second 
obversion, the last becomes 

(4) All birds are not-not-bipeds. 

As double negation destroys itself, this is equal to (i). 
Notice that the obverse of the obverse is the original. 

22. Give the converse of the contradictory of the 
proposition, There are no coins which are not 
made of metal. 

The premise is stated in a complex form with double 
negation; it means - No coins are not made of metal, 
which is the obverse of All coins are made of metal (A). 
The contradictory, as shown in the square of opposition 
(p. 31) is a proposition in O, namely, Some coins are 
not made of metal, which can be converted only by nega 
tion, that is, by joining the negative particle to the predicate, 
thus: Some coins are not -made -of- metal, whence by 
simple conversion Some things not-made-of-metal are 
coins the answer required. 



48 IMMEDIATE INFERENCE. [CHAP. 

23. (i) All crystals are solid. 

(2) Some solids are not crystals. 

(3) Some not-crystals are not solids. 

(4) No crystals are not-solids. 

(5) Some solids are crystals. 

(6) Some not-solids are not crystals. 

(7) All solids are crystals. 

Assign the logical relation, if any, between each 

of these propositions and the first of them. 
Proposition (i) is a universal affirmative (A); its simple 
obverse is (4); its converse by limitation is (5); the sub- 
contrary of this converse is (2). In order to obtain (6) 
we must take the contrapositive of (i), namely, All not- 
solids are not crystals/ the subaltern of which is (6); and 
converting (6) by negation we get (3). Again, (7) is the 
inverse, but is not inferrible from (i). We may further say 
that (4) can be inferred from (i), and is exactly equivalent 
in logical force to it ; (5) and (6) can be inferred, but are 
not equivalent to the original; (2) cannot be inferred from 
(i), but is not inconsistent with its truth. 

24. What information about the term not-^4 can 
we derive from the premise * All As are s ? 

This question, though apparently a very simple one, does 
not admit of a very simple answer ; it is important in a 
theoretical point of view. It may be said on the one hand, 
that as the proposition only affirms of all As that they are 
JBs, this tells us nothing about things excluded from the 
class A. Thus what is not-A may be B, or it may not be 
JB, without any interference from the premise. This is quite 
true. About Not- A universally we may infer nothing. 

But, on the other hand, if we convert the proposition all 
As are s by contraposition (p. 43), we get all not-^s 



v.] QUESTIONS AND ANSWERS. 49 

are not As. Uniting the negative particle to the predicate, 
we have All not-^s are not-^s, whence, by limited 
conversion, we infer some not-^s are not-^s. In this 
result we must interpret some as meaning, one at least, it 
may be more or even all. We shall recur to this question 
in a subsequent chapter. 

25. Assuming that no organic beings are devoid 
of carbon, what can we thence infer respectively 
about beings which are not organic, and things 
which are not devoid of carbon ? 

The premise No organic beings are devoid of carbon 
is a universal negative proposition, and does not directly 
give information about beings which are not organic, and 
beings which are not devoid of carbon. But, if we join the 
negative particle to the predicate, we get All organic 
beings are not-devoid-of-carbon, whence, by limited con 
version, Some things not devoid of carbon are organic, 
which answers the second part of the question. 

Again, converting by contraposition, we learn that All 
things not-not-devoid of carbon are not organic beings ; in 
other words, * All things devoid of carbon are not organic 
beings, a result which may be obtained perhaps more clearly 
by converting the original premise simply, thus, No things 
devoid of carbon are organic beings, or All things devoid 
of carbon are not organic beings. Conversion by limitation 
then yields Some things not organic beings are devoid of 
carbon, which is the answer to the first part of the question. 
This result is the same as that obtained in the last question, 
and the same remarks apply. 

26. What information about the term Solid Body 
can we derive from the proposition, No bodies 

which are not solids are crystals ? 
E 



50 IMMEDIATE INFERENCE. [CHAP. 

This question differs from the last only in being put in a 
more involved form. The premise when more simply 
stated becomes All not solids are not crystals, the contra- 
positive of All crystals are solids, and limited conversion 
gives Some solids are crystals. 

27. Nihil potest placere, quod non decet. Con 
vert this proposition, (i) simply, (2) by con 
traposition ; and show by what logical processes 
we can pass back from the contrapositive to 
the original. [c.] 

This premise (from Quinctilian, c. xi. 65) equals, Nihil quod 
non decet, potest placere; nothing which is unbecoming can 
please. Being a universal negative, E, it can be converted 
simply into Nothing which can please is unbecoming. 

In order to apply contraposition, we must put the pre 
mise into the form of A, thus All unbecoming things are 
unpleasing things, the contrapositive of which is All not 
unpleasing things are not unbecoming things, which having 
a double negative in each term equals All pleasing things 
are becoming. We can regain the original premise by 
applying contraposition to this last result. 

28. Convert, and give some immediate inferences 
from the following: Nothing is harmless that 
is mistaken for a virtue. 

The predicate of this proposition is clearly harmless/ 
and that is mistaken for a virtue is a relative clause 
describing the subject. The proposition is then Nothing 
mistaken for a virtue is harmless/ (E), converted simply 
into another proposition in E, Nothing harmless is mis 
taken for a virtue. 

Applying obversion to the original proposition we get 



v.] QUESTIONS AND ANSWERS. 51 

All that is mistaken for a virtue is not-harmless, or is 
harmful. By immediate inference by complex conception, 
we infer All foolish conduct mistaken for virtue is harmful 
foolish conduct. (Concerning inference by complex con 
ception, see Thomson s Outline, 88, and Elementary Lessons, 
P- 87.) 

29. Because every Prime Minister is a man, can 
we infer that every good Prime Minister is a 
good man ? 

The process of immediate inference by added deter 
minants, as described by Dr. Thomson, allows us to join an 
adjective or determining mark to both terms of an affirmative 
judgment, narrowing both terms, but to the same extent. 
Of course, however, it must be the same determining mark 
in each case, and if an adjective be ambiguous it is not 
logically the same adjective in its several meanings. Now 
good applied to a Prime Minister means that he is an able, 
active, upright minister, but probably very different from 
men who are good in other ranks of life. A good man 
means one who is good in the ordinary business and 
domestic relations of life. Thus the inference is erroneous. 
(See Elementary Lessons, p. 86.) It will afterwards be shown 
that when the proposition is fully expressed no such failure 
of inference can occur. Strictly speaking the premise is 

Prime Minister = Prime Minister, Man ; 
and it follows inevitably that 

Good, Prime Minister = Good, Prime Minister, Man. 

30. Euler employed two overlapping circles to 
represent a particular proposition. Can you 
raise any objection to the accuracy of such a 
diagram ? 



52 IMMEDIATE INFERENCE. [CHAP. 

Such circles have been employed in a great number of 
logical works. In my Elementary Lessons (p. 75) the 
particular proposition some metals are not brittle, is 
represented by the following figure : 




FIG. i. 

It does not seem to have been sufficiently noticed that 
though such a diagram correctly shows the exclusion of a 
part of the class metals from any part, that is all parts, of 
the class brittle substance, it indicates at the same time that 
another part of the class metals is included among brittle 
substances. Thus the diagram corresponds to the two pro 
positions I and O, instead of showing either apart from the 
other. Now, it has been fully explained that O is consistent 
with the truth of E ; so that when we say * some metals are 
not brittle, it may be that no metals are brittle, which is 
contradictory to I, some metals are brittle. The diagram 
should not prejudice this question, and it would therefore 
be best to remove the part of the circle bounding metals 
which falls within the circle of brittle substances, or else to 
have a broken line, as in Fig. 2. 




FIG. 2. 





v.] QUESTIONS AND ANSWERS. 53 

In the same way the proposition I, for instance, Some 
crystals are opaque/ would be represented by a broken circle 
included within a complete circle, in the manner shown 
either in Fig. 3 or Fig. 4. 

/ 3 

/ 2 
( I 

o 

FIG. 3. FIG. 4. 

31. What is the logical force of the following 
sentence from Sidgwick s Methods of Ethics : 
A materialist will naturally be a determinist; 
a determinist need not be a materialist ? 

Taking naturally to give a universal force to the first 
proposition, it becomes All materialists are determinists. 
The second proposition informs us that a determinist need 
not be a materialist, that is to say, at the least, some 
determinists are not materialists. This proposition is the 
sub-contrary of the converse of the first, and is the con 
tradictory of all determinists are materialists. The second 
proposition, then, prevents us from supposing materialists 
and determinists to be two co-extensive terms. We learn 
that there are persons called materialists who are all found 
among determinists; hence some called determinists are 
found among materialists ; other determinists, however, are 
not among materialists, and as to those who are not deter 
minists, they cannot be materialists. The first proposition 
would be technically described as A, and the second as O 
the contradictory of the inverse of the first. 



54 IMMEDIATE INFERENCE. [CHAP. 

32. All equilateral triangles are equiangular. 
May we thence infer that triangles having 
unequal angles have unequal sides, and vice 
versa f 

The proposition asserts that all equal-sided triangles have 
equal angles ; hence we may by contraposition infer that 
triangles which have not equal angles cannot have equal 
sides. But as the proposition stands, we are not justified in 
reading it reciprocally (see p. 32), and inferring that triangles 
which have not equal sides have not equal angles. This is 
true as a matter of geometrical science, but it is the contra- 
positive of another proposition, namely, the inverse all 
equiangular triangles are equilateral/ the truth of which 
must be separately proved. 

33. Can we ever convert a proposition of the 
form all As are >s into one of the form all 
s are As ? 

Certainly we cannot infer that all s are As because all 
As are JBs. As a general rule the predicate of the con- 
vertend B will be a wider term than the subject A, so that 
the inverse could not be inferred. Professor Henrici 
(Elementary Geometry, Congruent Figures, p. 14), for in 
stance, describes space as a three-way-spread, but we cannot 
convert simply, and say that every three-way-spread is space. 
It nevertheless happens not uncommonly that the original 
proposition is really intended to mean all As are all JBs, 
which can then be simply converted. Thus if space be 
defined as a three-way-spread of points, we can convert into 
every three-way-spread of points is space. Such definitions 
are of the form of proposition afterwards described by the 
symbol U (chapter xviii.), and considerable care is requisite 



v.] QUESTIONS AND ANSWERS. 55 

in discriminating between the propositions A and U. 
J. S. Mill has pointed to the simple conversion of a 
universal affirmative proposition as a very common form 
of error (System of Logic, book v., chapter vi., section 2). 
It cannot be too often repeated that the reciprocal and 
inverse propositions as described on p. 32, cannot be 
inferred from an original of the form A. 

34. In what cases does predication involve real 
existence ? Show that in some processes of 
conversion assumptions as to the existence of 
classes in nature have to be made ; and illus 
trate by examining whether any such assump 
tions, and if so what, are involved in the 
inference that if all 5 is P, therefore some 
not-S is not P. 

The above question must have been asked under some 
misapprehension. The inferences of formal logic have 
nothing whatever to do with real existence ; that is, occur 
rence under the conditions of time and space. No doubt, 
if all S is P, it follows that, in order to avoid logical con 
tradiction, some not-6 1 must be admitted to be not P. For 
instance, if All heathen gods are described in writings 
more than 1000 years old, it follows that Some things 
which are not heathen gods are not described in writings 
more than 1000 years old. This involves no assertion 
of real existence, nor could such an inference ever be 
drawn, unless, indeed, the original proposition itself asserted 
existence in time and space. This subject is pursued in 
a subsequent chapter. 



CHAPTER VI. 

EXERCISES ON PROPOSITIONS AND IMMEDIATE INFERENCE. 

1 . EXAMINE the following pairs of propositions, and decide 
which pairs contain consistent propositions, such that if the 
first of the pair be true the second may be true ; and vice 
versa^ if the second be true, the first may be true. Give 
the technical name of the logical relation, if any, between 
the two propositions of each pair. 

/ \ J Some metals are useful. 
I All metals are useful. 

. ^ ( No metals are useless. 

i Some useful things are not metals. 

Some useless things are metals. 
All useful things are metals. 

Some metals are useful. 
No metals are useless. 

All metals are useful. 

Some useless things are not metals. 

2. Draw all the immediate inferences you can from the 
proposition Knowledge is power. 

3. Give the converse of the contrapositive of the pro 
position All organic substances contain carbon. 



CHAP, vi.] EXERCISES ON PROPOSITIONS. 57 

4. Give all the logical opposites of intuta qua indecora, 
1 Unsafe are all things unbecoming. 

5. What information about the term * solid body can 
we derive from the proposition, No bodies which are not 
solids are crystals ? 

6. Only British subjects are native born Englishmen. 
What precisely does this proposition tell us about the four 
terms British subject. 

Not-British-subject. 
Native born Englishmen. 
Not-native-born-Englishmen ? 

7. Describe the logical relation between each of the 
four following propositions, and each of the other three : 

(1) All substances possess gravity which are material. 

(2) No substances which possess gravity are immaterial. 

(3) Some substances which are immaterial do not possess 

gravity. 

(4) Some substances which do not possess gravity are 

immaterial. 

8. State the nature and technical name of the logical 
process by which we get each of the following propositions 
from the preceding one : 

All men are mortal. 
No men are immortal. 
No immortals are men. 
None but mortals are men. 
All not-mortals are rot men. 
No men are not-mortals. 
All men are mortals. 

9. What are the subaltern propositions corresponding 
to the following universal propositions ? 



58 IMMEDIATE INFERENCE. [CHAP. 

(1) Every effect follows from a cause. 

(2) No one is admitted without payment. 

(3) All trespassers will be prosecuted with the utmost 

rigour of the law. 

(4) Nemo me impune lacessit. 

10. Give the obverse, converse, inverse, and reciprocal 
of each of the following propositions : 

(1) All mammalia are vertebrate animals. 

(2) Sir Rowland Hill is dead. 

(3) That which is a merit in an author is often a fault in 

a statesman. 

(4) Whatever is necessary exists. 

(5) In veritate victoria. 

11. Give the contrary, contradictory, subaltern, converse, 
obverse, inverse, contrapositive, and reciprocal propositions 
corresponding to each of the following propositions : 

(1) All B.A. s of the University of London have passed 

three examinations. 

(2) All men are sometimes thoughtless. 

(3) Uneasy lies the head which wears a crown. 

(4) The whole is greater than any of its parts. 

(5) None but solid bodies are crystals. 

(6) He who has been bitten by a serpent is afraid of a 

rope. 

(7) He who tries to say that which has never been said 

before him, will probably say that which will never 
be repeated after him. 

1 2. Give as many equivalent logical expressions as you 
can for the propositions 

(i) If the treasury was not full, the tax-gatherers were 
to blame. 



vi.] EXERCISES. 59 

(2) Through any three points not in a straight line a 

circle may be described. 

(3) It is false to say that only the virtuous prosper in 

life. [R.] 

13. What logical relations are there between the following 
propositions ? 

(1) All elementary substances are undecomposable. 

(2) There are no compounds which are not decom 

posable. 

(3) Some compounds are not decomposable. 

(4) No undecomposable substances are compounds. [E.] 

14. From the proposition Perfect happiness is im 
possible can we infer that Imperfect happiness is 
possible ? 

15. Is it the same thing to affirm the falsity of the pro 
position Some birds are predatory, and to affirm the truth 
of the proposition Some birds are not predatory ? 

1 6. Explain the statement that in the case of subcontrary 
propositions, truth may follow from falseness, but falseness 
cannot follow from truth. 

17. Give in succession (i) the obverse, (2) the converse, 
(3) the subaltern, (4) the contrary, (5) the contradictory, 
(6) the contrapositive of the proposition All wise acts are 
honest acts. 

1 8. Concerning the same proposition answer the following 
questions : 

(1) How is its converse related to its subaltern? 

(2) How is its converse related to the converse of its 

subaltern ? 

(3) How is its subaltern related to its contradictory? 

[BAGOT.] 



60 IMMEDIATE INFERENCE. [CHAP. 

19. What is the converse of the contrary of the con 
tradictory of the proposition Some crystals are cubes ? 
How is it related to the original proposition ? 

20. What is the converse of the converse of No men 
are ten feet high ? 

21. Name the logical process by which we pass from 
each of the following propositions to the succeeding 
one : 

(1) All metals are elements. 

(2) No metals are non-elements. 

(3) No non-elements are metals. 

(4) All non-elements are not metals. 

(5) All metals are elements. 

(6) Some elements are metals. 

(7) Some metals are elements. 

22. (i) None but a logical author is a truly scientific 
author. Taking this proposition as a premise, examine 
the following propositions, and decide which of them can 
be inferred from the premise. 

(2) A truly scientific author is no author who is not 

logical. 

(3) Some truly scientific authors are not any authors 

who are not logical. 

(4) A not truly scientific author is not a logical author. 

(5) Those who are not truly scientific authors cannot 

be logical. 

(6) All logical authors are truly scientific. 

(7) No truly scientific author is an illogical author. 

(8) All not illogical authors are truly scientific. 

(9) No illogical author is a truly scientific author. 

(10) No one is a truly scientific author who is not a logical 

author, 
(n) Some logical authors are not truly scientific authors. 



vi.] EXERCISES. 61 

Give, as far as possible, the technical name of the logical 
relation between each of the above propositions and each 
other. 

23. Some small sects are said to have no discreditable 
members, because they do not receive such, and extrude all 
who begin to verge upon the character. Point out how 
this statement illustrates logical conversion. 

24. Can we logically infer that because heat expands 
bodies, therefore cold contracts them ? 

25. Does it follow that because every city contains a 
cathedral, therefore the creation of a city involves the 
creation of a cathedral, or the creation of a cathedral in 
volves the creation of a city ? 

26. All English Dukes are members of the House of 
Lords. Does it follow by immediate inference by complex 
conception that the creation of an English Duke is the 
creation of a member of the House of Lords ? 

27. Give every possible converse of the following pro 
positions 

(1) Two straight lines cannot enclose space. 

(2) All trade-winds depend on heat. 

(3) Some students do not fail in anything. [M.] 

28. Give the logical opposites, converse and contra- 
positive, of Euclid s (so-called) twelfth axiom 

If a straight line meet two straight lines, so as to make 
the interior angles on the same side of it taken together less 
than two right angles, those straight lines being continually 
produced shall at length meet upon that side on which are 
the angles which are less than two right angles. 

29. How is the above proposition related to this other : 
If a straight line fall upon two parallel straight lines, it 



62 IMMEDIATE INFERENCE. [CHAP. 

makes the two interior angles upon the same side together 
equal to two right angles ? [R.] 

30. From Some members of Parliament are all the 
ministers (Elementary Lessons, p. 325, No. 3 [4]), can we 
infer that some place -seeking prejudiced and incapable 
members of Parliament are all the place-seeking prejudiced 
and incapable ministers ? 

31. Is it perfectly logical to argue that because two sub- 
contrary propositions may both be true at the same time, 
therefore their contradictories, which are contrary to each 
other, may both be false ? 

32. Is it perfectly logical to argue thus? If contrary 
propositions are both false, their respective contradictories, 
which are sub -contraries to each other, are both true. 
Now as this result is possible, it is therefore possible that 
the contraries may both be false. 

33. What is the logical relation, if any, between the 
two assertions in Proverbs, chap. xi. i, A false balance 
is abomination to the Lord: but a just weight is his 
delight ? 

34. Examine the verses of Proverbs, chap. x. to xv. 
and assign the relation between the two opposed assertions 
which make nearly all the verses. 

35. What is the nature of the step from anger is a short 
madness to madness is a long passion ? [R.] 

36. The angles at the base of an isosceles triangle are 
equal. What can be inferred from this proposition by 
obversion, conversion, and contraposition, without any 
appeal to geometrical proof? 

37. From the assertion The improbable is not impos 
sible, what can we learn, if anything, about (i) the 
possible, (2) the probable, (3) the not-improbable, (4) the 
impossible, (5) the not-impossible ? 



vi.] EXERCISES. 63 

38. How would a logician express the relations between 
the following statements of four interlocutors ? 

(1) None but traitors would do so base a deed. 

(2) And not all traitors. 

(3) Some would. 

(4) No ; not even traitors. 

[College Moral Science Examination, Cambridge.] 

39. What difficulties or absurdities do you meet in con 
verting the following propositions ? 

(1) Some books are dictionaries. 

(2) No triangle has one side equal to the sum of the 

other two. 

(3) Every one is the best judge of his own interests. 

(4) A few men are both scientific discoverers and men 

of business. 

(5) Whatever is, is right. 

(6) Some men are wise in their own conceit. 

(7) The eye sees not itself, 

But by reflection, by some other things. 



CHAPTER VII. 

DEFINITION AND DIVISION. 

i. ALMOST all text-books of Deductive Logic give rules 
for judging of the correctness of definitions, and for 
dividing up notions into subaltern genera and species. On 
attempting, however, to treat these parts of logic in the 
manner of this work, it has come home to me very strongly 
that they are beyond the sphere of Formal Logic, and 
involve the matter of thought. In form there is nothing 
peculiar to a definition ; in fact the very same proposition 
may be a definition to one person and a theorem to another. 
It is open to us for instance to define the number 9 as 
9 = 3 x 3; or, 9 = 8 + 1; or, 9 = 7 + 2, etc.; but having selected 
at will any one of these equations as a definition, the other 
equations follow as theorems. The perplexity in which the 
theory of parallel lines is involved partly arises from the 
fact that there is choice of definitions, some mathematicians 
choosing one way and some the other. It is quite apparent, 
too, that the same proposition may aftord different know 
ledge to different people. For instance, John Herschel 
was the only son of William Herschel would serve as a 
definition of John Herschel to any one who knew only 
William Herschel, and of William Herschel to one who 
only knew John. To one who knew both it might be a 



CHAP, vii.] EXERCISES. 65 

theorem. Similar remarks might be made concerning the 
distinction between ampliative and explicative propositions. 

2. These in addition to other considerations convince me 
that any attempt to treat definition as a part of Formal 
Logic must be theoretically unsound and practically un 
satisfactory. The case is somewhat similar with Logical 
Division, which, so far as it belongs to Formal Logic, can 
be nothing more than that method of Dichotomous Division 
fully developed in the later chapters on Equational Logic. 
Anything more than this must involve material knowledge, 
and should be treated in a different work and in a different 
manner. On these grounds I have decided not to attempt 
any explication of Definition and Division here, but to con 
fine this chapter to a collection of questions, such as are to 
be commonly found in examination papers. The student 
may be referred for the current doctrines to the Elementary 
Lessons ; Nos. XII. and XIII.; Fowlers Deductive Logic, 
Chapters VII. and VIII. ; Duncarfs Logic, Chapter VI. etc. 

3. Examine the following definitions 

(1) Conversion is the changing of terms in a proposi 

tion. 

(2) Opposed propositions are those which differ in 

quantity and quality. 

(3) Contradictory opposition is the opposition of con 

tradictories. [R.] 

4. Define any of the following terms, notions, or classes 
of objects 

Gravitation Franchise Communism 

Consistency Imagination Honour 

Library Honesty Club 

Vegetable Revenge Dictionary 



66 DEFINITION AND DIVISION. [CHAP. 

Diet Syllogism Conservative 

Hypochondriac Racehorse University 

Merit Success Specie. 

5. Criticise the following definitions 

(1) A square is a four -sided figure of which the sides 

are all equal and the angles all right angles. 

(2) A member of the solar system is anything over which 

the sun has continued influence. 

(3) La vie est le contraire de la mort. 

(4) A lemma is a proposition which is only used as 

subservient to the proof of another proposition. 

(5) An archdeacon is one who exercises archidiaconal 

functions. 

(6) Life is the definite combination of heterogeneous 

changes, both simultaneous and successive, in 
correspondence with external coexistences and 
sequences. 

(7) A gentleman is a man having no visible means of 

subsistence. [ORTON.] 

(8) Equal bodies are those whereof every one can fill 

the place of every other. [HOBBES.] 

6. Examine the definitions 

(1) Tin is a metal lighter than gold. 

(2) Vice is the opposite of virtue. 

(3) Paper is a substance made of rags. 

(4) Cheese is a caseous preparation of milk. 

(5) Rust is the red desquamation of old iron. 

(6) A transcendental function is any function which is 

not an algebraic function. 

(7) A right-angled triangle is a triangle containing one 

right angle, and of which the containing sides 
are or are not equal. 



vii.] EXERCISES. 67 

(8) An organ is any part of an animal or plant appro 

priated to a distinct function. 

(9) A man is a self-knowing animal. 

(10) Knowledge is that on which somebody else can be 

examined. [ROLLESTON.] 

( 1 1 ) An animal is a sentient organised being. 

(12) A triangle is a three-sided figure having its angles 

together equal to two right angles. 

(13) A man is one who may be the Prince of Tran 

sylvania. [HOBBES.] 

7. In what respects are the following definitions, or some 
of them, defective ? 

(1) Logic is a guide to correct reasoning. 

(2) Logic is the art of expressing thoughts in correct 

language. 

(3) Logic is a mental science. 

(4) Logic is the science of the regulative laws of human 

thought. 

8. Does the eleventh chapter of the Hebrews, or any 
part of it, contain a correct logical definition of Faith ? 

9. Give examples of indefinable words, and explain why 
words may be indefinable. 

10. Give the Proximate Genera for the following 
species 

Man Plant Monarchy 

Triangle Hound Science. 

1 1. Define by genus and differentia the following terms ; 
and name a proprium and an accident in each case : 

Island Parallelogram 

Bank Bill of exchange 

Dictionary Tree. 



68 DEFINITION AND DIVISION. [CHAP. 

1 2. What are the genus, species, difference, property, and 
accident of Examination ? [D.] 

13. Distinguish specific attribute, property, and accident, 
using the term Circle as an example. [B.] 

14. Is it possible to define the terms gold, coal, legal 
nuisance, civilisation, Cleopatra s needle? 

15. Define the term boat, and then point out how many 
of the following things the definition includes : Bark, 
ferry-boat, floating fire-engine, pontoon, wherry, canoe. 

1 6. Classify the following objects under one or other of 
the heads, cash, bills, specie : 

Cheque, promissory note, shilling, money, token -coin, 
bank-note, I.O.U., paper -money, sovereign, Scotch bank 
note. (See Money and the Mechanism of Exchange (Inter 
national Scientific Series), p. 248. Section on the Definition 
of Money.) 

17. Distinguish Logical from Physical Division and 
Definition. [o.] 

1 8. Can anything admit of more than one defini 
tion ? [o.] 

19. Distinguish precisely between the definition and the 
description of a class. 

20. Explain the difficulties which arise concerning the 
definition of parallel straight lines, and criticise the following 
suggested definitions : 

(1) Lines which are in every part equidistant. 

(2) Lines of similar direction. 

(3) Lines which being in the same plane and indefinitely 

prolonged never meet. 



vii.] EXERCISES. 69 

21. Examine the following definitions : 

(1) Man is a bundle of habits. 

(2) Law is common sense. 

(3) Reverence is the feeling which accompanies the 

recognition of superiority or worth in others. 

(4) Hunger is the product of man s reflection on the 

necessity of food. [p.] 

22. Which of the following are logical divisions, and 
which are not ? 

(1) Man into civilised and uncivilised. 

(2) The world into Asia, America, Europe, Africa, 

Australia. 

(3) Grammar into syntax and prosody. 

(4) War into civil and aggressive. 

(5) Syllogisms into those which are logical and illogical. 

(6) Sequences of phenomena into casual and causal. 

(7) Energy into potential and visible. 

(8) Geometrical figures into plane and tri-dimensional. 

(9) Allegiance is either natural and perpetual, or local 

and temporary. 

23. Divide the term Inference, so as to include the 
various species usually discussed by logicians. [E.] 

24. The following were the classes of persons which were 
in 1868 qualified to vote in one or other of the United 
States of America : Male citizen, male inhabitant, every 
man, white male citizen, white freeman, male person, white 
male adult, free white male citizen, free white man. 

Form a scheme of logical division which shall have a 
place for each of the above classes. 

25. Divide logically University, colour, chair, science, 



70 DEFINITION AND DIVISION. [CHAP. vn. 

religion, species, art, church, undergraduate, virtue, states 
man, [o.] 

26. Form a scheme of division of sciences to include the 
species Deductive, experimental, concrete, descriptive, 
rational, abstract, inductive, explanatory, empirical. 

27. Apply the rules of logical division to the following 
instances, correcting what is wrong, and supplying what is 
deficient : 

(1) Discursive thought may be divided into the Term, 

Judgment and Syllogism. 

(2) Notions are Concrete, Singular, and Universal. 

(3) Propositions are Affirmative, Negative, and Universal. 

28. To what extent are the rules of division, usually 
given in logical treatises, repudiated by the classifications 
adopted in the Natural Sciences ? [L.] 

29. When is a division inadequate? When indistinct? 
When a cross division ? And when not arranged according 
to proximate parts ? [MORELL.] 

30. Give an accurate scheme of logical division in which 
the following things shall find places: Name; Part of 
Speech ; Term ; Vox logica ; Verb ; Noun, Adjective ; 
Syncategorematic term ; Word. 



CHAPTER VIII. 

SYLLOGISM. 

i. MEDIATE Inference, or Syllogism, forms the principal 
part of Deductive Logic, and offers a wide scope for useful 
exercises. I give, in the first place, a brief epitome of the 
syllogistic rules and forms ; I then exemplify them abun 
dantly by question and answer ; lastly, I supply chapters 
full of the largest and most varied collection of syllogistic 
questions and problems which has ever been published. 
Some of the more perplexing questions, involving the dis 
tinction of formal and material falsity of syllogisms and 
their premises, are treated apart in the succeeding 
chapter (xii.). 

RULES OF THE SYLLOGISM. 

1 i ) Every syllogism has three and only three terms. 
These terms are called the major term, the minor term, 

and the middle term. 

(2) Every syllogism contains three and only three proposi 
tions. 

These propositions are called respectively the major 
premise, the minor premise, and the conclusion. 

(3) The middle term must be distributed once at least. 

(4) No term must be distributed in the conclusion which 
was not distributed in one of the premises. 



72 SYLLOGISM. [CHAP. 

(5) From negative premises nothing can be inferred. 

(6) If one premise be negative, the conclusion must be nega 
tive ; and vice versa, to prove a negative conclusion one of the 
premises must be negative. 

From the above rules may be deduced two subordinate 
rules, which it will nevertheless be convenient to state at 
once. 

(7) From two particular premises no conclusion can be 
drawn. 

(8) If one premise be particular, the conclusion must be 
particular. 

FIGURES OF THE SYLLOGISM. 

S = minor term. M = middle term. P = major term. 

First Figure. Second Figure. Third Figure. Fourth Figure. 



M...P P...M M...P P...M 
Minor S ... M S ... M M ... S M ... S 



Premise. 

Minor 
Premise. 



Conclusion. S...P S...P S...-T b...x 

MOODS OF THE SYLLOGISM. 

The following is a compact table of the valid moods of 
the syllogism, the numerals showing the figures in which 
each group of propositions makes a valid syllogism : 

AAA AAI AEE All AGO 

i. 3-4- 2.4. 1.3. 2. 

EAE EAO EIO IAI OAO 
1.2. 3.4- 1.2.3.4. 3.4. 3. 

MNEMONIC VERSES. 

Barbara, Celarent, Darii, Ferioque, rmoris ; 
Cesare, Camestres, Festino, Baroko, secundae ; 



viii.] QUESTIONS AND ANSWERS. 73 

Tertia, Darapti, Disamis, Datisi, Felapton, 
Bokarclo, Ferison, habet ; Quarts insuper addit, 
Bramantip, Camenes, Dimaris, Fesapo, Fresison. 

Certain letters in the above lines indicate the way in 
which the moods of the second, third, and fourth figures 
may be reduced to the first figure, as follows 

s directs you to convert simply the proposition denoted 

by the preceding vowel. 
p directs you to convert the proposition per accident, or 

by limitation. 

m, for muta, directs you to transpose the premises. 
k denotes that the mood can only be reduced per 

impossibile. 

The initial consonant of each mood in the three last 
figures corresponds with the initial of the mood of the first 
figure to which it is reducible. 



QUESTIONS AND ANSWERS. 

2. State the figure and mood to which the sub 
joined argument belongs : 

Iron is not a compound substance; for iron 
is a metal, and no metals are compounds. 

The conjunction * for shows that the proposition pre 
ceding is the conclusion a universal negative. The term 
* metal must be the middle term, because it does not ap 
pear in the conclusion. The major term being compound 
substance, the major premise must be * no metals are com 
pound substances, (E) and the other premise iron is a 
metal must be the minor. The latter is a universal affirm- 



74 SYLLOGISM. [CHAP. 

ative ; for though no mark of quantity is prefixed to iron, 
it states a chemical truth concerning iron in general, and may 
fairly be interpreted universally (A). The argument belongs 
to the mood EAE in the first figure, or Celarent, thus : 

E No metals are compound substances. 

A (all) Iron is a metal. 

E Iron is not a compound substance. 

3. Examine the following argument ; throw it 
into a syllogistic form, and bring out the 
figure and mood : 

It cannot be true that all repression is mis 
chievous, if government is repressive and yet 
is sometimes beneficial. [B.] 

The conclusion is stated in the form of a denial of the 
universal affirmative all repression is mischievous ; hence 
the contradictory of this, or some repression is not mis 
chievous is the real conclusion. The middle term is 
government/ which does not appear in the conclusion. 
In looking for the major term, we do not find mischievous 
in the premises, but only its opposite term beneficial. 
We must assume, then, that we are intended to take bene 
ficial as equivalent to not-mischievous, otherwise there 
would be a fallacy of four terms. To be brief, then, the 
syllogism takes this form 

Some government is not mischievous. 

All government is repressive (or repression). 

Therefore, some repression is not mischievous. It is a 
valid syllogism in the third figure, and mood OAO, or 
Bokardo. 



VIII.} QUESTIONS AND ANSWERS. 75 

4. In what figures is the mood AEE valid ? 

In the first figure we have 

All M is P. 
No S is M. 
No S is P. 

The negative conclusion distributes the major term P, 
which is undistributed in the major premise ; hence Illicit 
Process of the Major Term. 

In the second figure we have 

All P is M. 
No S is M. 
No S is P. 

The major term is now properly distributed in the major 
premise, and the middle term being also distributed once, 
in the minor negative premise, the syllogism is valid in 
Camestres. 

The reader may show that in the third figure we have 
again Illicit Process of the Major, and in the fourth figure 
a valid syllogism Camenes. 

5. What rules of the syllogism are broken by 
arguments in the pseudo-moods, OAE, and 
OIE? 

The answer cannot be better given than in the words of 
Solly (Syllabus of Logic, p. 86). In the mood OAE the 
predicate is distributed in the major premise, and the 
subject in the minor premise, and both subject and predi 
cate in the conclusion. Hence it follows that either some 
term must be distributed in the conclusion which was not 
distributed in the premises, or else the middle term cannot 



76 SYLLOGISM. [CHAP. 

be distributed in either premise. We cannot, therefore, 
determine at once which form the fallacy will take, but may 
be quite certain that there must be either an illicit process 
of major or minor, or else an undistributed middle. 

Again, in the mood OIE, both subject and predicate 
are distributed in the conclusion, whereas no term is distri 
buted in the minor premise, and it therefore follows that 
there must be an illicit process of the minor. It is also 
evident that the middle term cannot be distributed in 
the minor premise, and that if it is distributed in the 
major premise the major term must be undistributed, and 
consequently there must be a fallacy either of undistributed 
middle or illicit major. 

6. None but whites are civilised ; the ancient 
Germans were whites : therefore they were 
civilised. [w.] 

This appears at first sight to be in Barbara, the terms 
standing apparently in the order of the first figure. But the 
major premise does not assert that all whites are civilised ; 
it only asserts that none but whites are so, and this is equiva 
lent to the contrapositive of the proposition 

All civilised are whites. 
Joining to this the minor premise 

The ancient Germans were whites, 

we see that the argument is in the second figure, with two 
affirmative premises, so that the middle term is undistri 
buted in both cases, producing Fallacy of Undistributed 
Middle. There is also a difference of tense between the 
two premises which might perhaps invalidate an argument ; 
but this point need not be further noticed here. 



vin.] QUESTIONS AND ANSWERS. 77 

7. None but civilised people are whites ; the 
Gauls were whites : therefore they were 
civilised. [w.] 

At first sight this seems to be in the second figure, and 
invalid; but converting the major premise by contraposition, 
as in the last example, we get a valid syllogism in Barbara 
thus, All whites are civilised; the Gauls were whites, etc. 

8. All books of literature are subject to error ; 
and they are all of man s invention ; hence 
all things of man s invention are subject to 
error. [H.] 

This may seem at the first reading to be correct reason 
ing, especially as the conclusion is materially true ; but 
there is fallacy of Illicit Process of the Minor Term. The 
argument is in the pseudo-mood AAA of the third figure, 
and the conclusion should be * some things of man s invention 
are subject to error. 

9. He who is content with what he has is truly 
rich ; a covetous man is not content with what 
he has ; no covetous man, therefore, is truly 
rich. 

The middle term is content with what he has, and since 
this term appears as subject of the major premise and 
predicate of the minor, the argument is in the first figure in 
the pseudo-mood AEE. There is Illicit Process of the 
Major Term, because the conclusion E distributes its 
predicate and the major premise A does not. 

The conclusion may be true in matter but does not follow 
from the premises. We could only make the argument good 



7 8 SYLLOGISM. [CHAP. 

by taking as major premise, All the truly rich are content 
with what they have. This would give a valid syllogism in 
Camestres, but the original premise, if converted, only 
yields Some truly rich are content with what they have. 

10. Protection from punishment is plainly due to 
the innocent ; therefore, as you maintain that 
this person ought not to be punished, it appears 
that you are convinced of his innocence, [w.] 

The above is equivalent to 

The innocent are not to be punished ; 
This person is not to be punished ; 
Therefore, this person is innocent. 

Put in this form there is an obvious fallacy of Negative 
Premises ; but we can also express the premises in an 
affirmative form as follows : 

The innocent ought to be exempt from punishment ; 
This person ought to be exempt from punishment. 
In this case it is apparent that the middle term ought to be 
exempt from punishment is undistributed in both the 
premises, against Rule 3 of the syllogism. 

11. He that is of God heareth my words : ye 
therefore hear them not, because ye are not 
of God. [w.] 

In the usual order : 

He that is of God heareth my words ; 

Ye are not of God ; 
. . Ye do not hear my words. 
The propositions are AEE in the first figure, and involve 



viii.] QUESTIONS AND ANSWERS. 79 

the Fallacy of Illicit Process of the Major Term. Hear 
my words is distributed as the predicate of the negative 
conclusion, but is undistributed as the predicate of the 
affirmative major premise. The argument would become 
valid, however, if we were allowed to quantify this predicate 
universally, and assume the meaning to be 

He that is of God = he who heareth my words. 



12. Any books conveying important truths with 
out error deserve attention ; but as such books 
are few, it is plain that few books do deserve 
attention. 

Carefully distinguish the truth and fallacy in 
this argument. 

A good example suggested by Whately s No. 13 ; it may 
be thus put : 

Any books conveying, etc., deserve attention ; 
Few books do convey, etc. ; 
. . Few books do deserve attention. 

This is in the first figure, and, if we interpret the con 
clusion to mean that ^ A few books do deserve attention, 
that is to say, affirmatively only, without implying that the 
rest do not, it is valid in the mood Darii. But usually 
(Elementary Lessons, p. 67), we interpret/^/ negatively; 
indeed, in the example itself this is the plain meaning, such 
books are few, implying that all but this few do not convey 
important truths without error. This makes the minor pre 
mise into O, Most books do not, etc., and the argument 
consisting of AOO in the first figure is a case of Illicit 
Process of the Major Term. 



8o SYLLOGISM. [CHAP. 

13. That man is independent of the caprices of 
Fortune, who places his chief happiness in 
moral and intellectual excellence : A true 
philosopher is independent of the caprices of 
Fortune : therefore a true philosopher is one 
who places his chief happiness in moral and 
intellectual excellence. [w.] 

A case of the fallacy of Undistributed Middle, the middle 
term independent of the caprices of Fortune, being pre 
dicated affirmatively both of one who places his chief 
happiness, etc., and of a true philosopher. The pseudo- 
mood is, therefore, A A A in the second figure. The fallacy 
is none the better because the conclusion may be considered 
true in matter. If the premise had begun Only that man 
is independent, etc., we might have put the argument into 
a valid syllogism, Barbara. 

14. It is an intensely cold climate that is sufficient 
to freeze quicksilver ; and as the climate of 
Siberia does this it is intensely cold. 

At the first glance this looks like a case of Undistributed 
Middle ; but we soon see that the major premise is really 
Any climate sufficiently cold to freeze quicksilver is an 
intensely cold climate. The argument is thus valid in 
Barbara. 

15. No one who lives with another on terms of 
confidence is justified, on any pretence, in 
killing him : Brutus lived on terms of confi 
dence with Caesar : therefore he was not justi 
fied, on the pretence he pleaded, in killing 
him. [w.] 



viii.] QUESTIONS AND ANSWERS. 81 

This is valid in Barbara, the major term being * justified on 
any pretence, etc., the middle term, one who lives, etc., 
and the minor term, Brutus. 

The conclusion, however, is obviously weakened, or is 
less general than it might have been. We might conclude 
that Brutus was not justified in killing Caesar on any pretence. 
It is only inferred that he was not justified in killing him on 
the pretence he pleaded, which is of course included in any 
pretence. 

16. Inquire into the validity of the following 
argument : Whatever substance is properly 
called by the name Coal consists of a carbon 
aceous substance found in the earth ; now, as 
this specimen consists of a carbonaceous sub 
stance, and was found in the earth, therefore it 
is properly called Coal. [L.] 

The above argument is evidently a case of Undistri 
buted Middle, because we infer that this specimen is 
properly called Coal on the ground of two universal affirma 
tive propositions, both of which have the same predicate 
consisting of a carbonaceous substance found in the earth. 
The pseudo-mood then is AAA in the second figure. 

Though entirely failing in a demonstrative point of view, 
it is another question whether the specimen may not be 
believed to be coal, on analogical or inductive inference. 

17. Give any remarks which occur to you con 
cerning the following : Nerve power does not 
seem to be identical with electricity; for it is 
found that when a nerve is tightly compressed 
nervous action does not go on, but electricity 
can nevertheless pass/ 

G 



82 SYLLOGISM. [CHAP. 

Implies the following syllogism : 

All tightly compressed nerves do not convey nervous 

action ; 

All tightly compressed nerves do convey electricity ; 
Therefore, some things which convey electricity do not 

convey nervous action. 

The propositions are AAI in the third figure, i.e. the 
syllogism is valid in Darapti. It is matter of further 
inference that, because electricity is conducted by some 
things which do not convey nervous action, therefore these 
actions are not identical. 

18. * With some of them God was not well 
pleased ; for they were overthrown in the 
wilderness. [w.] 

An enthymeme of the first order, the major premise 
being omitted. The order of statement is that by some 
logicians called analytical, the conclusion being put first, 
and the minor premise adduced as a reason or proof. The 
major premise, assumed to be obvious, is to the effect that 
* All who are overthrown in the wilderness are among those 
with whom God is not well pleased. More fully stated, 
indeed, the assumption might be that all who suffer from 
any signal calamity are some of those with whom God is 
not well pleased. To be overthrown in the wilderness is to 
suffer from a signal calamity. This view makes a sorites 
which the reader can put in order. 

Ip. If the major term of a syllogism be the 
predicate of the major premise, what do we 
know about the minor premise ? [L.] 

In answering syllogistic questions of this sort, great 



VIIL] QUESTIONS AND ANSWERS. 83 

attention must be given to throwing the reasoning into the 
briefest and clearest form. Such questions, thus treated, 
afford capital exercises in reasoning. The above question 
may be answered thus : 

If the major premise is affirmative its predicate, the major 
term, is undistributed and must likewise be undistributed in 
the conclusion; in this case the conclusion, and consequently 
the minor premise, must be affirmative. If the major pre 
mise be negative, then the minor premise must be affirmative, 
in order to avoid negative premises ; thus in any case the 
minor premise is affirmative. Or still more briefly thus : 

The minor must be affirmative, for if negative then the 
major would have to be affirmative, which would involve 
Illicit Process of the Major. 

20. Prove that O cannot be a premise in the 
first or fourth figure ; and that it cannot be 
the major in the second figure, or the minor 
in the third. [M.] 

If either of the premises be O, the conclusion must 
be negative, so that its predicate the major term will be 
distributed. But as O distributes only its predicate, and 
the other premise, which must of course be affirmative and 
universal, only distributes its subject, the syllogistic conditions 
are much restricted. Thus OA as premises in the first 
figure give an Undistributed Middle, the middle term being 
subject of O and predicate of A. The premises AO give 
Illicit Process of the Major Term, the predicate of A, the 
major term, being undistributed. In the second figure O 
cannot be the major, because its subject would then be the 
major term, and undistributed. In the third figure it 
cannot be the minor, because the major term would then 



84 SYLLOGISM. [CHAP. 

be predicate of A, the major premise, and thus Illicit Pro 
cess of the Major would again arise. Finally, in the fourth 
figure OA will give Illicit Major, and AO, Undistributed 
Middle. 

21. If it be known concerning a syllogism in the 
Aristotelian system, that the middle term is 
distributed in both premises, what can we infer 
as to the conclusion ? 

The syllogism cannot be in the second figure, because the 
middle term, being the predicate in both premises, these 
would both have to be negative, against Rule 5. In the 
first figure the minor premise would have to be negative, in 
order to distribute its predicate, the middle term ; but a 
negative minor in the first figure gives Illicit Process of the 
Major Term. In the third figure, however, the middle term 
being subject of both premises will be twice distributed if 
these be both universal, which happens in the moods Darapti 
and Felapton. In the fourth figure the middle term is pre 
dicate of the major and subject of the minor ; we must, 
therefore, have a negative major and a universal affirmative 
minor, which happens in the mood Fesapo. We find, then, 
that a doubly distributed middle term can prove only a 
particular conclusion, I or O, and these only in the third 
and fourth figures. 

22. Take an apparent syllogism subject to the 
fallacy of negative premises, and inquire 
whether you can correct the reasoning by 
converting one or both of the premises into 
the affirmative form. 

[India Civil Service, July 1879.] 



QUESTIONS AND ANSWERS. 85 

Take premises in the first figure 

No Fis Z; 
No X is Y. 

Obvert the major premise (see p. 42), and we have- 
All Y is not-Z; 
No X is F. 

The premises would give no conclusion, the pseudo-mood 
AEE in the first figure involving Illicit Process of the 
Major. Obverting the minor, we have 

No Fis Z; 
All Xis not-F. 

There are now four terms, and therefore no common 
middle term at all. The reader may easily work out other 
examples. (See Principles of Science, p. 62 ; first edition, 
Vol. I. p. 75-) 

23. Prove that the third figure must have an 
affirmative minor premise, and a particular 
conclusion. 

In the third figure the major term is predicate of the 
major premise. Now, if the minor premise be negative, 
the conclusion will be negative (Rule 6), and distribute its 
predicate the major term, but the major premise must be 
affirmative in order to avoid negative premises. Thus, there 
will arise Illicit Process of the Major Term. It follows, by 
rednctio ad absurdum, that the minor premise cannot be 
negative and must be affirmative. 

Again the predicate of the minor premise is the minor 
term, and, the premise being affirmative, this term will be 
undistributed, giving a particular conclusion. 



86 SYLLOGISM. [CHAP. 

24. Show that if the conclusion of a syllogism be 
a universal proposition, the middle term can 
be but once distributed in the premises. 

Questions of this sort can be most briefly answered by 
counting the available number of distributed terms in the 
premises. Thus, if the conclusion be a universal affirma 
tive proposition, we need one distributed term for its 
subject. But, as the premises must both be affirmative, 
they contain at most two distributed terms, namely their 
subjects. Hence there is only one place in which the 
middle term can be distributed. On the other hand, if the 
universal conclusion be negative, both major and minor 
terms require to be distributed in the premises; but, as one 
premise only can be negative, we cannot possibly have more 
than three terms distributed, the subject and predicate of 
the negative premise, and the subject of the affirmative one. 
Two being required for the major and minor terms, there 
remains only one distributed place for the middle term, 
which was to be proved. 

Observe that this result is the contrapositive of that 
proved under Question 21 (p. 83). 

25. Given the six rules of the syllogism, and 
the rule that two particular premises prove 
nothing, show that if one premise be par 
ticular the conclusion must be particular. 

This may be demonstrated by the following ingenious 
reasoning of De Morgan (Formal Logic, p. 14). 

If two propositions, P and Q, together prove a third, , 
it is plain that P and the denial of R prove the denial of 
Q. For P and Q cannot be true together without ./?. 



vin.] QUESTIONS AND ANSWERS. 87 

Now, if possible, let P (a particular) and Q (a universal) 
prove fi. (a universal). Then P (particular) and the denial 
of R (particular) prove the denial of Q. But two parti 
culars can prove nothing. 

26. Show that the proposition O is seldom 
admissible as a minor premise. 

When O is the minor premise the conclusion must be 
negative by Rule 6, and will therefore distribute its pre 
dicate, the major term. As we must not have two negative 
premises by Rule 5, the major premise must be affirmative, 
and will not distribute its predicate. Hence the major 
term must be the subject of the major premise. Now, 
since the middle term becomes the undistributed predicate 
of the major premise, it must be the predicate of the minor 
premise, in order that it may be once distributed. Thus 
we conclude that O can be the minor premise only in the 
second figure, giving the mood Baroko. 

27. Show that a universal negative proposition 
(E) is highly efficient as a major premise. 

M 

Since E has both its terms distributed, either of them 
may serve as the major term, which, the conclusion being 
negative, must be distributed. The other term will then 
serve to distribute the middle term once. The minor pre 
mise may therefore be chosen at will, provided that it be 
affirmative in order to avoid negative premises. There is 
no restriction of figure, and accordingly we find valid moods 
with E as major premise in all of the four figures ; in fact, 
no less than eight of the nineteen recognised moods begin 
with E. 



88 SYLLOGISM. [CHAP. 

28. Name the weakened moods of the syllogism. 
In what figure can there be no weakened 
mood, and why ? Do any of the nineteen 
moods commonly recognised give a weaker 
conclusion than the premises would warrant ? 

By a weakened mood is meant one which gives a particular 
conclusion when a universal conclusion might have been 
drawn. The information obtained from the premises is 
thus weakened. This can, of course, happen only when 
the conclusion of the stronger mood is universal. Hence, 
in the third figure, which gives only particular conclusions, 
there can be no weakened mood. In the other figures each 
mood which has a universal conclusion will have a corre 
sponding weakened mood with conclusion of the same 
quality. Thus Barbara gives a mood A A I ; Celarent, 
EAO; Cesare, EAO; Camestres, AEO; in the fourth 
figure only Camenes admits of a weakened form, AEO. 
Thus the weakened moods are five in number. 

Bramantip of the fourth figure is the single mood alluded 
to in the latter part of the question. 

Considering that it is impossible to employ conversion 
by limitation without weakening the logical force of the 
premise, it is too bad of the Aristotelian logicians to slight 
the weakened moods of the syllogism as they have usually 
done. 

29. Can we under any circumstances infer a 
relation between X and Z from the pre 
mises 

Some Fs are Xs ; 
Some Fs are Zs ? 

[India Civil Service, July 1879.] 



viii.] QUESTIONS AND ANSWERS. 89 

Not if some Fs bear the sense attributed to the ex 
pression in Logic. The indefinite adjective of quantity 
some is so indefinite, that it must never be interpreted twice 
over with the same meaning. But if the some Y in the 
one premise were intended by the arguer to be the same 
some V as in the other premise, the term would practically 
become a distributed one, and the premises might give a 
valid conclusion in the mood Darapti. Dr. Thomson has 
remarked (Laws of Thought, 77, p. 132), that the word 
(some) appears to be employed in the two senses of 
" Some or other," and "Some certain," in common language. 
Observe, however, that it is in the former purely indefinite 
sense that logicians have always used the word, so that 
some K* must not be identified with some K 

30. Is the following argument a valid syllogism ? 
That which has no parts cannot perish by the 
dissolution of its parts ; the soul has no 
parts ; therefore, the soul cannot perish by 
the dissolution of its parts. 

This example is quoted from the Port Royal Logic, Part 
III. Chap, ix., Example 6. It is there remarked that 
several persons advance such syllogisms in order to show 
the inaccuracy of the unconditional rule (5) that nothing 
can be inferred from negative premises. Without re 
membering what was said in the Art of Thinking, I made 
the same objection in the Principles of Science, p. 63 (first 
ed. Vol. I. p. 76), and I must still hold that in its bare 
statement the syllogistic rule is actually falsified. But it 
must, no doubt, be allowed that if the premises are to 
be treated as both negative, then there are four terms ; the 
middle term is broken up into two terms, that which has 



90 SYLLOGISM. [CHAP. 

not parts/ and that which has parts ; the soul is denied 
to be the latter ; the former is that of which it is asserted 
negatively that it cannot perish, etc. It comes simply to 
this, that the syllogistic rules are to be interpreted as a 
whole, and in making the above example conform to the 
first rule (see p. 71) we make it conform also to the fifth 
rule. 

Professor Groom Robertson has criticised my treatment 
of this subject (Mind, 1876, p. 218, note}, urging that 
1 There are four terms in the example, and thus no syllogism, 
if the premises are taken as negative propositions ; while 
the minor premise is an affirmative proposition, if the terms 
are made of the requisite number three. No doubt 
Professor Robertson is substantially right, but it may be 
noticed that my words were so cautious as hardly to commit 
me to an erroneous statement. I now find that the point 
has been treated by many logicians in addition to those of 
Port Royal, as for instance, Burgersdicius ; De Morgan, 
Formal Logic, p. 139, Art. 3; Bain, Deductive Logic, p. 164; 
Devey, Logic, or the Science of Inference, 1854, p. 129; 
Essai sur la Logique, 1763, p. 106. 

31. In reference to the syllogism, Mr. Jevons 
urges that it sometimes yields a conclusion 
that is open to misinterpretation, as in the 
example- 
Potassium is a metal ; 
Potassium floats on water ; 
Therefore, Some metal floats on water. 

Examine this criticism carefully. 

[Moral Science Tripos, Cambridge, Dec. 1876.] 
I said in the Principles of Science (pp. 59-60; first ed. 



QUESTIONS AND ANSWERS. 91 

Vol. I. pp. 71-2), that my inference, namely, Potassium 
metal = potassium floating on water, is of a more exact 
character than the Aristotelian result Some metal floats on 
water. The some after all is only here an indefinite 
name for potassium, and unless we constantly bear in 
mind that some means in logic one, and it may be more 
or all, the reasoner is apt to confuse some with the 
plural several. This view of the matter was criticised by 
Professor Groom Robertson (Mind, 1876, p. 219). 

32. What is the nature of the argument, if any, 
in the apparent enthymeme, The field is 
neglected because the soil is poor ? 

This may, of course, be an argument in Barbara, thus 
Every field of poor soil is neglected ; 
This is a field of poor soil ; 
. . This field is neglected. 

But the statement may also mean that the soil being poor 
is the reason or cause why the owner neglects it ; in this 
case, it is not an argument but a causal relation. The 
student, therefore, must always look out for ambiguities in 
the conjunction for, because, etc., which may certainly 
bear one of two if not of more senses. The relation between 
premises and conclusion has nothing whatever to do with 
the relation between cause and effect. 

33. Explain It is scarcely ever possible de 
cidedly to affirm that an} argument involves 
a bad syllogism ; but this detracts nothing 
from the value of the syllogistic rules. [R.] 

Scarcely any one in ordinary writing or discourse states a 
syllogism in full form ; it is always presumed that the hearer 



92 SYLLOGISM. [CHAP. 

or reader is enough of a logician to supply what is wanting. 
Now the missing premise may generally be supplied in such 
a way as to make a good syllogism formally speaking, that 
is to say, so as to avoid any breach of the syllogistic rules. 
It is another matter whether the new premise is materially 
true. The value of the syllogistic rules is, then, that they 
enable us to assign the premises which would be requisite 
to support the conclusion put forward. They thus oblige 
the arguer to define the nature of his assumptions, or else 
to yield up his conclusion. 

34. How shall we reduce the following syllogism 
to the first figure ? 

All men are liable to err ; 

None who are liable to err should refuse advice : 
None who should refuse advice are men. 
This argument is in the absurd fourth figure, in the mood 
Camenes. In this name the letter m directs us to transpose 
the premises, and the final s directs us to convert the con 
clusion simply ; making these changes, we obtain the same 
argument in the more natural form of Celarent, thus 
None who are liable to err should refuse advice ; 

. -.*. 
All men are liable to err ; 

No men should refuse advice. ^ 

35. How shall we reduce the following syllogism 
to the first figure ? 

All birds are vertebrates ; 
Some winged animals are not vertebrates ; 
Some winged animals are not birds. 
The premises are A O in the second figure, and the con 
clusion being O, the argument is a valid syllogism in Baroko. 



VIIT.] QUESTIONS AND ANSWERS. 93 

The letter k directs us to employ the Reductio ad 
impossibile, as explained in the Elementary Lessons, p. 149. 
Or we may convert the major by contraposition, getting 

All not-vertebrates are not birds ; 
Some winged animals are not-vertebrates ; 
Therefore, Some winged animals are not birds. 

Taking the negative term not-vertebrates as the middle 
term, this is valid in Barbara. 



36. Can we reduce the mood Camestres per 

impossibile ? 
Taking the symbolic example 

All Xs are Ys ; 
No Zs are Fs ; 
Therefore, No Zs are Xs, 

and assuming for sake of argument that the conclusion is 
false, the contradictory * some Zs are Xs will be true, which 
put as minor premise with the original as major, gives the 
valid syllogism in Darii 

All ^s are Fs ; 

Some Zs are Xs ; 
Therefore, Some Zs are Fs. 

But this conclusion is the contradictory of the original 
minor premise no Zs are Fs, so that we cannot contradict 
the conclusion of Camestres without producing a syllogism 
in Darii to contradict one of our original premises. Thus 
we prove the conclusion of Camestres indirectly by a mood 
of the first figure. It will be found on trial that all the 
moods of the imperfect figures may be similarly proved 
indirectly by one or other of the moods of the first or 
so-called perfect figure, 



CHAPTER IX. 

QUESTIONS AND EXERCISES ON THE SYLLOGISM. 

i. ASSIGN the moods of the following valid syllogisms, 
pointing out in succession 

(a) The conclusion ; 

(b) The middle term ; 

(c) The major term, and the major premise containing it ; 

(d) The minor term, and the minor premise containing it ; 

(e) The quantities and qualities of the three propositions ; 
(/) Their symbols ; 

(g) The order in which they should be technically placed ; 

(h) The figure of the syllogism ; 

(/) The mood, and its mnemonic name. 

(1) No birds are viviparous ; 

All feathered animals are birds ; 
No feathered animals are viviparous. 

(2) Robinson is plain spoken ; for he is a Yorkshire man, 

and all Yorkshire men are plain spoken. 

(3) Birds are not viviparous animals ; 
Bats are viviparous animals ; 
Bats, therefore, are not birds. 

(4) Whatever investigates natural laws is a science ; 
Logic investigates natural laws ; 

| Logic is a science. 



CHAP, ix.] QUESTIONS AND EXERCISES. 95 

(5) Quicksilver is liquid at ordinary temperatures ; 
Quicksilver is a metal ; 

Some metal, therefore, is liquid at ordinary tempera 
tures. 

(6) True fishes respire water containing air ; 
Whales do not respire water containing air ; 
Whales, therefore, are not true fishes. 

2. Arrange the following valid syllogisms in the usual 
strict order of major premise, minor premise and conclusion. 
Name the figure and mood to which they belong. In exa 
mining syllogisms, always follow the directions of the first 
question. 

(1) Iridium must be lustrous; for it is a metal, and all 

metals are lustrous. 

(2) Some pleasures are not praiseworthy; hence some 

pleasures are not virtuous, for whatever is not 
praiseworthy is not virtuous. 

(3) Epicureans do not hold that virtue is the chief good, 

but all true philosophers do hold that it is so; 
accordingly, epicureans are not true philosophers. 

(4) Some towns in Lancashire are unhealthy, because 

they are badly drained, and such towns are all 
unhealthy. 

3. Draw conclusions from the following pairs of premises, 
specifying the figure and mood employed 

,,( Every virtue is accompanied with discretion; 

1 There is a zeal without discretion. 
. ^ f Sodium is a metal ; 

1 Sodium is not a very dense substance. 
All lions are carnivorous animals ; 
No carnivorous animals are devoid of claws, 



96 SYLLOGISM. [CHAP. 

( Combustion is chemical union ; 

(4) -] Combustion is always accompanied by evolution 

(. of heat. 

C All boys in the third form learn algebra ; 

(5) \ There are no boys in the third form under twelve 
V years of age. 

Nihil erat quod non tetigit : 
Nihil quod tetigit non ornavitj 

4. Examine the following arguments and point out which 
are valid syllogisms, naming the figure and mood as before ; 
in the case of such as are pseudo-syllogisms, name the rule 
of the syllogism which is broken thereby, and give the 
technical name of the fallacy 

(1) All feathered animals are vertebrates ; 
No reptiles are feathered animals ; 
Some reptiles are not vertebrates. 

(2) Some vertebrates are bipeds ; 
Some bipeds are birds ; 
Some birds are vertebrates. 

(3) All vices are reprehensible ; 
Emulation is not reprehensible ; 
Emulation is not a vice. 

(4) All vices are reprehensible ; 
Emulation is not a vice ; 
Emulation is not reprehensible. [L.] 

(5) Some works of art are useful ; 

All works of man are works of art ; 
Therefore some works of man are useful [L.] 

(6) Iron is a metal ; 

All metals are soluble ; 
Iron is soluble. 



ix.] QUESTIONS AND EXERCISES. 97 

(7) Aryans are destined to possess the world ; 
Chinese are not Aryans ; 

Chinese are not destined to possess the woWd. 

(8) Only ten-pound householders have votes ; 
Smith is a ten-pound householder ; 
Smith has a vote. 



5. What are the suppressed premises which are evidently 
presumed to exist by those who set forth the following 
imperfectly stated syllogisms? State figure and mood as 
usual. 

(1) Blessed are the meek : for they shall inherit the earth. 

(2) This iron is not malleable ; for it is cast iron. 

(3) Whosoever loveth wine shall not be trusted of any 

man ; for he cannot keep a secret. 

(4) Being born in Africa, he was naturally black. 

(5) Some parallelograms are not regular plane figures, for 

they cannot be inscribed in a circle. 

(6) Suffer little children to come unto me ; for of them is 

the kingdom of Heaven. 

(7) It is dangerous to tell people that the laws are not 

just ; for they only obey laws because they think 
them just. 

(8) The line A B is equal to the line CD; for they are 

both radii of the same circle. 

(9) Whales are not true fishes, for they respire air; more 

over they suckle their young. 
(10) The Queen is at Windsor, for the royal standard is 

flying, 
(n) The science of logic is very useful; it enables us to 

detect our adversaries fallacies. 
(12) He must be in York, for he is not in London. 



98 SYLLOGISM. [CHAT. 

(13) I shall not derive my opinions from books for I 

have none. [Mansfield, H. of L., 1780.] 

(14) The nation has a right to good government; there 

fore it may rebel against bad governors. 

(15) The wise man has an infinity of pleasures ; for virtue 

has its delights in the midst of the severities that 
attend it. 

6. Point out which of the following pairs of premises will 
give a syllogistic conclusion, and name the obstacle which 
exists in other cases. 

(1) No A is B ; some B is not C. 

(2) No A is B \ some not C is B. 

(3) All B is not A ; some not A is B. 

(4) Some not A is B ; no C is B. 

(5) All not B is C; some not A is B. 

(6) All A is B \ all not C is not B. 

(7) All not B is not C all not A is not B. 

(8) All ^ is not ^ ; no B is not C. 

(9) All C is not B ; no ^ is not ./?. 

7. To what moods do the following belong? 

(1) < All ^ is A ; only C is A ; therefore only C is B: 

(2) l All B is ^4 ; nothing but C is ^ ; therefore nothing 

but Cvs. B: 

See Dante s De Monarchia, as translated by F. C. Church, 
and appended to the Essay on Dante, by the Rev. R. W. 
Church, 1878, p. 195. Many curious specimens of reason 
ing, sometimes pedantic, might be drawn from the De 
Monarchia. 

8. Supply premises to prove or disprove the following 
conclusions 



ix.] QUESTIONS AND EXERCISES. 99 

(1) The loss of the Captain proves that turret-ships are 

not sea-worthy. 

(2) The cottage-hospital system should be adopted. 

(3) The Prussians are justified in refusing the rights of 

war to Garibaldi if they find him fighting against 
them. [E.] 

(4) Private property should be respected in war. 

(5) No woman ought to be admitted to the franchise, [o.] 

(6) The law of libel requires to be amended. [o.] 

(7) Capital punishment ought to be abolished. [o.] 

(8) Royal parks ought not to be used for political meet 

ings, [o.] 

(9) Written examinations are not a safe test of merit. 
(10) Written examinations are a safe test of merit. [E.] 
(n) The Annuity-tax should be done away with. [E.] 
(12) Any national system of education should be a secular 

system. [E.] 

9. In how many different moods may the argument im 
plied in the following question be stated ? No one can 
maintain that all persecution is justifiable who admits that 
persecution is sometimes ineffective. 

How would the formal correctness of the reasoning be 
affected by reading deny for * maintain ? [c.] 

i o. What conclusions, and of what mood and figure, can 
be drawn from each pair of the following propositions ? 

(1) None but gentlemen are members of the club. 

(2) Some members of the club are not officers. 

(3) All officers are invited to dine. 

(4) All members of the club are invited to dine, [c.] 

ii. Express the following reasonings in each of the four 
syllogistic figures. 



ioo SYLLOGISM. [CHAP. 

(1) Some medicines should not be sold without registering 

the buyer s name, for they are poisons. [E.] 

(2) No unwise man can be trusted ; hence some specula 

tive men are unworthy of trust, for they are 
unwise. [E.] 

12. Can the following argument be stated in the form of 
a syllogism, and if so, what is the middle term ? 

The power of ridicule is a dangerous faculty, since it 
tempts its possessor to find fault unjustly, and to distress 
some for the gratification of others. 

13. If the proposition warmth is essential to growth 
occurred as the premise of a syllogism, would you treat 
warmth as a distributed or an undistributed term ? [E.] 

14. Show that the following single propositions may be 
regarded as enthymemes, that is as equivalent to imperfectly 
expressed syllogisms : 

(1) Have thou nothing to do with that just man. [w.] 

(2) If wishes were horses, beggars would ride. 

(3) Large colonies are as detrimental to the power of a 

State, as overgrown limbs to the vigour of the 
human body. 

(4) If I had read as much as my neighbours, I would 

have been as ignorant. [HOBBES.] 

(5) All law is an abridgment of liberty and consequently 

of happiness. 

(6) Thales being asked what was the most universally 

enjoyed of all things, answered Hope ; for they 
have it who have nothing else. 

(7) I will give thee my daughter if thou canst touch 

heaven. 



ix.] QUESTIONS AND EXERCISES. 101 

(8) If all the absurd theories of lawyers and divines were 
to vitiate the objects in which they are conversant, 
we should have no law and no religion left in the 
world. [BURKE.] 

15. Distinguish between the causal, simply logical, or 
other, senses of the copulative conjunctions in the 
following 

(1) It will certainly rain, for the sky looks black. 

(2) The people are happy because the government is 

good. 

(3) This plant is not a rose ; for it is monopetalous. 

(4) The ancient Romans trusted their soothsayers, and 

must therefore have been frequently deceived. 

(5) A favourable state of the exchanges will lead to im 

portation of gold : this will cause a corresponding 
issue of bank-notes which will occasion an ad 
vance in prices ; which again will check exportation 
and encourage importation, tending to turn the 
exchanges against us. [GILBART, 1851, p. 284.] 

1 6. Form an example of a syllogism in which there are 
two prosyllogisms, one attached to the middle and the 
other to the minor term. [H.] 

1 7. Prove that a valid sorites with n premises must have 
;/ + i terms, and is capable of giving n ^ n ~ * conclusions. 

2 

1 8. Can the following Shakspearean passage (Hamlet, 
Act v. Scene i.) be stated in the form of a sorites ? 

Alexander died, Alexander was buried, Alexander 
returneth into dust ; the dust is earth ; of earth we make 
loam ; and why of that loam, whereto he was converted, 
might they not stop a beer barrel? 



102 SYLLOGISM. [CHAP. ix. 

19. Throw the reasoning of the following passage into 
syllogistic form : 

Carbon, which is one of the main sources of the 
nourishment of plants, cannot be dissolved in water in its 
simple form, and cannot therefore be absorbed in that form 
by plants, since the cells absorb only dissolved substances. 
All the carbon found in plants must consequently have 
entered them in a form soluble in water, and this we find in 
carbonic acid, which consists of carbon and oxygen. [A.] 

20. Complete such of the following arguments as may be 
considered sound but incomplete syllogisms : 

(1) The people of the country are suffering from famine, 

and as you are one of the people of the country, 
you must be suffering from famine. 

(2) Light cannot consist of material particles, for it does 

not possess momentum. 

(3) Aristotle must have been a man of extraordinary 

industry; for he could not otherwise have pro 
duced so many works. 

(4) Marcus Aurelius was both a good man and an 

Emperor ; hence it follows that Emperors may be 
good men, and vice versa. 

(5) Nothing which is unattainable without labour is 

valuable; some knowledge is not attainable with 
labour, and is therefore valuable. 

(6) All gasteropods are mollusks, and no vertebrate 

animals are mollusks ; therefore no gasteropods 
are vertebrate. 

(7) Suicide is not always to be condemned ; for it is but 

voluntary death, and voluntary death has been 
gladly embraced by many great heroes. 



CHAPTER X. 

TECHNICAL EXERCISES IN THE SYLLOGISM. 

1. PROVE, from the general rules of Syllogism, that when 
the major term is predicate in its premise, the minor 
premise must be affirmative. 

2. Prove that,, when the minor term is predicate in its 
premise, the conclusion cannot be a universal affirmative. 

>] 

3. Prove that there must always be in the premises one 
distributed term more than in the conclusion. 

4. Prove that the major premise of a syllogism, whose 
conclusion is negative, can never be a particular affirmative. 

5. Prove that when the minor premise is universal nega 
tive, the conclusion (unless weakened) will be universal. 

6. Prove that, if in the first figure we transpose the major 
premise and conclusion, we obtain a pseudo-mood. 

7. In the third figure, if the conclusion be substituted for 
the major premise, what will the figure be ? [BAGOT.] 

8. Prove that no syllogism in the fourth figure can be 
correct which has a particular negative among its premises, 
or a universal affirmative for its conclusion. [L.] 

9. If the major term be universal in the premises and 
particular in the conclusion, determine the mood and figure, 
it being understood that the conclusion is not a weakened 
one. [c.] 



io 4 SYLLOGISM. [CHAP. 

10. Why is it impossible to transform the mood A E O 
from the second figure into the first ? 

1 1. What figure must have a negative conclusion ? Why 
must it? 

12. What figure must have a particular conclusion? 
Why must it ? 

13. Why must the major premise of the fourth figure not 
be O? 

14. Why must the minor premise of the fourth figure 
not be O ? 

15. If the minor premise of the first figure were not 
affirmative, what fallacy would be committed ? 

1 6. If the major premise of the first figure were I, what 
fallacy would be committed ? 

17. What kind of proposition does not occur in the 
premises of the first figure ? Why does it not occur ? 

1 8. If the major premise of the second figure were 
particular, what fallacy would be committed ? 

19. What is remarkable about the conclusions of the 
third figure ? 

20. What kind of proposition cannot be proved by the 
fourth figure? 

21. In what mood of the syllogism can a subaltern pro 
position be substituted for its subalternans (universal of 
same quality) as premise without affecting the conclusion ? 

22. If one premise of a syllogism be O, what must the 
conclusion be ? 

23. Prove that a universal affirmative proposition can 
form the conclusion of the first figure alone. 

24. Why is O A O excluded from the first and second 
figures ? 

25. Why is it that the moods E A O and E I O are true 
in all the four figures ? 



x.] TECHNICAL EXERCISES. 105 

26. It is said that A is the most difficult conclusion to 
establish by the syllogism, and the most easy to overthrow ; 
O, on the contrary, is the most easy to establish, and the 
most difficult to overthrow. What is there in the moods of 
the syllogism to support this view ? 

27. When both premises of an apparent syllogism are 
negative, the real middle term is an external sphere, and is 
consequently undistributed [SOLLY]. Explain the meaning 
of this statement. 

2 8. If the minor premise of a syllogism be O, what is the 
figure and mood? 

2 9. Prove that there cannot be more than four figures of 
the syllogism. 

30. If one premise be O, what must the other be? 

31. Show that in the fourth figure the conclusion may 
be either affirmative or negative, and, if negative, either 
universal or particular. 

32. Given the major premise particular, or the minor 
premise negative, what must the other premise be ? Why 
so? [P.] 

33. If the minor term be predicate of minor, or major 
subject of major, the conclusion may not be A? [p.] 

34. Prove that the combination of a particular major 
premise with a negative minor premise leads to no valid 
inference. 

35. Prove that wherever there is a particular conclu 
sion without a particular premise, something superfluous is 
invariably assumed in the premises. 

36. Determine in what affirmative moods the middle term 
may be universal in the major premise, and particular in the 
minor. 

37. Determine in what negative moods the same may 
occur. [pi 



io6 SYLLOGISM. [CHAP. x. 

38. Determine how many universal terms may be in the 
premises more than in the conclusion. 

39. Determine how many particular terms may be in the 
premises more than in the conclusion. 

40. Determine in what cases there may be in a syllogism 
an equal number of universal terms and of particular, [p.] 

41. How do you reduce Camestres, Festino, Darapti, 
Fresison to the first figure ? 

42. Exemplify the reduction of Baroko and Bokardo by 
the process per impossibile. 

43. Show that Cesare, Disamis, Camenes, and other 
moods can likewise be proved per impossibile. (See Kars- 
lake, 1851, Vol. I. p. 81.) 

44. Reduce Celarent to the fourth figure. To how 
many other figures can you reduce it ? 

45. Reduce Felapton to the second figure. 

46. To what other moods, respectively, can you reduce 
Darii, Ferio, Barbara? 



CHAPTER XL 

CUNYNGHAME S SYLLOGISTIC CARDS. 

1. IN this age of mechanical progress it may be a matter 
of surprise that no one has produced a syllogistic machine. 
About two centuries ago Pascal and Leibnitz invented true 
calculating machines, and Swift, incited perhaps by the 
accounts of these machines, described the professors of 
Laputa as in the possession of a thinking machine. About 
thirty years ago the late Alfred Smee, F.R.S., proposed the 
construction of a kind of mechanical dictionary, together 
with a contrivance for comparing the ideas defined in it. 
More recently I have constructed a machine which analyses 
the meaning of any propositions worked upon its keys, 
provided they do not involve more than four distinct terms. 
Yet the rules of the syllogism have never been put into a 
mechanical form. So much the worse for the rules of the 
syllogism. 

2. Some approximation to a syllogistic machine has, 
however, been recently made by Mr. Henry Cunynghame, 
B.A. He has devised certain cards, which, if placed one 
upon the other, infallibly give a syllogistic mood when 
that is possible, and when it is not, indicate the absence 
of a conclusion. The contents of the cards, too, can be 
condensed into a hollow cylinder turning upon a solid 
cylinder, in such a way as to give all possible syllogistic 



io8 SYLLOGISTIC CARDS. [CHAP. 

moods in one turn of the handle. This device, though 
hardly perhaps to be called a syllogistic machine, is probably 
the nearest approximation to such a machine which is pos 
sible. I am now enabled, by Mr. Cunynghame s kindness, 
to describe these ingenious and interesting devices for the 
first time. 

3. The syllogistic cards consist of a set of eight larger 
and a set of eight smaller cards. Each larger card is 3^ 
inches high by 2\ inches broad, and bears, near to its upper 
edge, one of the eight possible propositions connecting M, 
the middle term, with />, the major term. There are, of 
course, four propositions of the forms A, E, I, O, in which 
M is the subject, and four more in which P is the subject. 
In the lower part of each larger card is written, in a certain 
position, each proposition between S and P, which can 
possibly be drawn from a syllogism having the proposition 
on the upper edge of the card for its major premise. Thus, 
under the major premise * Some M is P] we find Some S 
is P, as the only conclusion that can be drawn from it ; 
but the universal affirmative proposition, All P is J/, 
admits of any one of three possible conclusions, namely, 
No S is P; Some S is P, or some S is not P. 1 

4. Each card of the lesser set is also 2-J- inches wide, but 
only 3 inches high, and bears on its upper edge one of the 
eight possible minor premises connecting S, the minor term, 
with M, the middle term. In the lower part of each smaller, 
or, as we may call it, minor card (with one exception), is cut 
one, or in some cases two rectangular openings, so adjusted 
that if any minor card be placed upon a larger or major 
card, so that the major and minor premises inscribed upon 
them be visible, one below the other, the conclusion, if any, 
belonging to those premises is seen through the opening in 
the minor card. If two conclusions, a universal proposition, 



XL] SYLLOGISTIC CARDS. 109 

and its subaltern minor, are both possible, both will be seen. 
Thus, if we take the major Some P is MJ and the minor 
All M is SJ we see below the conclusion * Some S is P, 1 
the mood being Dimaris in the fourth figure. If we take 
the minor All S is J/, and place it upon the major No 
M is P, we see below No S is P, the correct conclusion 
in the mood Celarent, together with Some S is not P, the 
corresponding weakened or subaltern conclusion. If the 
major be Some P is not M, or the minor Some M is 
not 5, no conclusion can appear at all. 

5. The cards are shown in complete detail on the next 
page. The principle on which they are constructed is that 
of excluding illogical conclusions. No conclusion is written 
on the major card but such as the premise at the top will 
warrant, and no conclusion is left uncovered by the minor 
card if it be unwarranted by the minor premise. The 
student can readily cut out cards according to the directions 
given above and the figures on the next page, but they 
must be cut exactly to scale. It will assist the con 
struction if each major card be divided by a pencil mark 
into seven horizontal spaces, each half an inch high, each 
minor card being divided into six similar spaces. 

6. The syllogistic cylinder is more compact and handy, 
but much less easy to describe and illustrate. The prin 
ciple is exactly the same, but considerable ingenuity was 
needed to combine the cards together into cylinders in the 
best way. The order of the minor premises on the in 
ferior or moving cylinder is as follows : All S is MJ 
1 All M is S, Some S is M; Some M is S t > No S is M, 
No M is SJ Some S is not MJ Some M is not 
The major premises are in like order, All P is M, All 

; Some -Pis M; etc. 



CUNYNGHAME S SYLLOGISTIC CARDS. 
MAJOR CARDS. 



All M is P 
All S is P 

Some S is P 




No M is P 
NoSisP 

Some 6" is not F 




Some M is P 
Some S is P 




Some ^f is not P 
Some 6" is not P 








All P is M 

No 6 1 is P 
Some S is P 
Some vS" is not P 


No /> is M 
NoSisP 

Some vS" is not P 




Some PisM 
Some S is /> 




Some /> is not M 


MINOR CARDS. 


All S is ^/ 




No 6" is M 




Some 6" is M 




So me 6" is not M 






















All J/ is 6 1 




No Af is 6 




Some M is 6" 




Some Af is not ,5" 














L 



CHAPTER XII. 

FORMAL AND MATERIAL TRUTH AND FALSITY. 

i. THE rules of syllogistic inference teach us how, 
from certain premises assumed to be true, to draw other 
propositions which will be true under those assumptions. 
But if, instead of supposing the premises to be true, we 
regard them as materially false, various puzzling questions 
arise as to the conclusions which may properly be drawn. 
Such questions have not been adequately treated in any 
popular manual of logic and as they lead to results of 
great practical importance, and at the same time furnish 
admirable exercises in the discrimination of good and bad 
reasoning, I propose to draw special attention to this 
subject in the following questions with answers. 

2. Is it possible to draw a false conclusion from 
true premises ? 

It is possible, of course, to draw any conclusion from any 
premises, if we disregard the principles of logic and of 
common sense. But when we speak in a logical work of 
drawing a conclusion, we must be understood to mean 
drawing a conclusion logically, in accordance with the Laws 
of Thought and the Rules of the Syllogism. Now the nature 
of the logical relation between premises and conclusion is 



ii2 TRUTH AND FALSITY. [CHAP. 

this, that if the premises are true the conclusion is true ; 
truth is, as it were, carried from the premises into the 
conclusion ; not the whole truth necessarily, but nothing 
except truth. The question above must, of course, be 
answered with a direct negative. 

3. Is it possible to prove a true conclusion with 
false premises ? 

To prove a true conclusion, or to prove that a certain 
conclusion is true, must mean to establish its truth in 
the opinion of the persons concerned. To prove, 
says Wesley, 1832, p. 90, is to adduce premises which 
establish the truth of some conclusion. Now, the rela 
tion of premises and conclusion in a syllogism, as stated 
just above, is that if the premises are true the conclusion 
must be admitted to be likewise true. But, if certain 
persons regard the premises as false, they cannot possibly 
regard such premises as establishing or proving the truth of 
the conclusion. Solly, indeed, points out (1839, note, p. 9) 
the possibility of a true conclusion from false premises in 
every form of reasoning. But this remark can only mean 
that the conclusion is materially true, or known to be true, 
on other grounds. 

4. If the premises of a syllogism are false, does 
this make the reasoning false ? 

No. The reasoning is correct if the form of the pre 
mises and conclusion agree with that of any valid mood of 
the syllogism or other development of the Laws of Thought, 
wholly regardless of the material truth of any of the propo 
sitions per se. The most ridiculous proposition may make 
a good syllogism : for instance 



xii.] QUESTIONS AND ANSWERS. 113 

Every griffin has angles equal to two right angles; 

Every triangle is a griffin ; 
Therefore, Every triangle has, etc. 

This is, of course, a valid syllogism in Barbara, and if the 
premises were true the conclusion would be true ; the pre 
mises being untrue, the truth of the conclusion is entirely 
unaffected by the reasoning. 

5. (a) The most perfect logic will not serve a 
man who starts from a false premise. 
(//) I am enough of a logician to know that 
from false premises it is impossible to draw a 
true conclusion. 
Comment carefully upon the two foregoing 

extracts. 

Both the above sentences have been written in perfect 
seriousness by men of intelligence, and they are fair speci 
mens of the logic which would pass muster almost anywhere 
except in a book of logical exercises. In the first place, as 
regards (a), if a premise be materially false it cannot give 
a conclusion materially true ; accidentally, indeed, it might 
do so by paralogism ; but, as the logic is assumed to be 
most perfect, we are dealing only with material truth and 
falsity. Nevertheless, the logic may serve the man well ; 
for he can learn the truth or falsity of his propositions 
by observing their congruity with external facts. Now, if 
he has failed to learn in this way directly the falsity of his 
premise, his only chance is to draw logical conclusions from 
that premise, and then observe whether they are or are not 
materially verified. It is quite clear that if by correct logic 
we reach a conclusion materially false, then we must have 
started from premises which involved material error. This 



1 1 4 TRUTH AND FALSITY. [CHAP. 

procedure represents, in fact, the real method of induction, 
as the inverse process of deduction , by which we learn all the 
more complicated truths of physical and moral science. (See 
Principles of Science, Chaps. XI. and XII). The sentence 
(a\ then, is true only on the supposition that a man, having 
adopted a false premise, will blindly accept all its false 
results, that is to say, will reason in a purely deductive 
manner. 

The sentence (b) is erroneous, because, as we have fully 
learnt (p. 112), we can from false premises draw a true 
conclusion in good logical form. But there was doubtless 
confusion in the writer s mind between formal and material 
falseness, and had he said that by premise^ materially false 
it is impossible to establish the material truth of any con 
clusion, he would have been correct. 

6. An apparent syllogism of the second figure 
being examined is found to break the rules of 
the syllogism, the middle term being undis 
tributed. On further examination it is re 
marked that one of the premises is evidently 
false, and the other true. What can we infer 
from such circumstances concerning the truth 
or falsity of the conclusion ? [c.] 

As in the second figure the middle term is predicate in 
both the premises, the apparent syllogism can break the 
third rule of the syllogism, requiring that it shall be distri 
buted once at least, only when both the premises are affirma 
tive. The premises must therefore be A A, A I, I A, or II. 
In the first case, if the premises be 
All ^s are Fs, 
All Zs are Fs, 



xii.] QUESTIONS AND ANSWERS. 115 

and we assume the first one to be false, we obtain its contra 
dictory as true ; thus 

Some Xs are not Fs ; 

All Zs are Fs. 

The conclusion must, by Rule 6, be negative, and there 
will be Illicit Process of the Major. Assuming the second 
premise to be false, we get 

All Xs are Fs ; 
Some Zs are not Fs. 

Whence we may correctly infer in the mood Baroko, 
Some Zs are not Xs. 

In the case of A I, we obviously cannot assume A to be 
false ; but if I be false, we get 

All Xs are Fs ; 

No Zs are Fs ; 

Therefore, No Zs are Xs. 

In the premises I A, we cannot assume A to be false without 
Illicit Process of the Major Term ; but if I be false, we 
have Cesare. Lastly, in 1 1 only the major can be taken 
as false, and its contradictory then gives us a syllogism in 
Festino. If the conclusion of the apparent or pseudo- 
syllogism in question does not correspond with what we 
thus obtain, the conclusion is logically false as compared 
with the new premises assumed. 

7. If (i) it is false that whenever X is found F 
is found with it, and (2) not less untrue that 
X is sometimes found without the accompani 
ment of Z, are you justified in denying that 
(3) whenever Z is found there also you may 



n6 TRUTH AND FALSITY. [CHAP. 

be sure of finding Y? And however this may 
be, can you in the same circumstances judge 
anything about Y in terms of Z ? [R.] 

This excellent example of reasoning by contradictories 
can be easily solved by adhering to the simple rules of 
opposition, and gradually undoing the perplexities. The 
supposition that, whenever X is found Y is found with it, 
may be stated as the universal affirmative all Xs are Ys, ; 
but as this is false, its contradictory some Xs are not Ys 
is the true condition. That (2) X is sometimes found with 
out the accompaniment of Z, would mean that some Xs 
are not Zs ; but, being asserted to be untrue, the real 
condition is its contradictory All Xs are Zs. Thirdly, 
whenever Z is found, there also you may be sure 01 rinding 
y, means that all Zs are ys ; but, if you deny this, you 
must assert that some Zs are not ys. Putting these 
propositions together, thus 

1 i ) Some Xs are not ys ; 

(2) A\\Xs areZs; 
Hence, (3) Some Zs are not ys ; 

we find that they make a valid syllogism in the third 
figure, and the mood Bokardo. The conclusion, being a 
particular negative, cannot be converted directly ; we can 
only obtain by obversion and conversion some not-ys 
are Xs. Thus we must, I presume, answer the last part 
of the problem negatively. 

8. What is the precise meaning of the assertion 
that a proposition say All grasses are edible 
is false ? 
The doctrine of the falsity of propositions is generally 



xii.] QUESTIONS AND ANSWERS. 117 

supposed to be defined with precision in the ancient formula 
of the square of opposition. If a universal affirmative pro 
position is false, its contradictory, the particular negative, is 
true, so that, in the case of the example given above, we 
infer, that some grasses are not edible. Similarly, from 
the falsity of E we infer the truth of I, and vice versa. 
But it does not seem to have occurred to logicians in general 
to inquire how far similar relations could be detected in the 
case of disjunctive and other more complicated kinds of 
propositions. Take, for instance, the assertion that All 
endogens are all parallel-leaved plants. If this be false, 
what is true ? Apparently that one or more endogens are 
not parallel-leaved plants, as, else that one or more parallel- 
leaved plants are not endogens. But it may also happen 
that no endogen is a parallel-leaved plant at all. There are 
three alternatives, and the simple falsity of the original does 
not show which of the possible contradictories is true. 

But the question arises whether there is not confusion 
of ideas in the usual treatment of this ancient doctrine of 
opposition, and whether a contradictory of a proposition is 
not any proposition which involves the falsity of the original, 
but is not the sole condition of it. I apprehend that any 
assertion is false which is made without sufficient grounds. 
It is false to assert that the hidden side of the moon is 
covered with mountains, not because we can prove the con 
tradictory, but because we know that the assertor must have 
made the assertion without evidence. If a person ignorant 
of mathematics were to assert that all involutes are tran 
scendental curves, he would be making a false assertion, 
because, whether they are so or not, he cannot know it. 
Professor F. W. Newman has correctly remarked that no 
one can really believe a proposition the terms of which he 
does not understand (Lectures on Logic, 1838, pp. 35, 36). 



n8 TRUTH AND FALSITY. [CHAP. 

This is unquestionably true; for, if he does not know 
what things he is speaking about, he cannot possibly bring 
them to comparison in his mind. A witness who swears that 
a prisoner did a certain act when, as a matter of fact, he 
does not know whether the prisoner did it or not, swears 
falsely, independently of the question whether rebutting 
evidence can be brought to prove the perjury. It is reported 
that a man, who wished to be thought an acquaintance of 
Dr. Johnson, remarked to him in coming out of church, A 
good sermon to-day, Dr. Johnson. That may be, sir, 
replied the very much over-estimated doctor, but I m not 
sure that you can know it. This hits the point precisely. 

It will be shown in a subsequent chapter that a pro 
position of moderate complexity has an almost unlimited 
number of contradictory propositions, which are more or 
less in conflict with the original. The truth of any one or 
more of these contradictories establishes the falsity of the 
original, but the falsity of the original does not establish the 
truth of any one or more of its contradictories, because 
there always remains the alternative that nothing is known 
concerning the relations of the terms. It may even happen 
that no relation at all exists between the terms. In this 
view of the matter, then, an assertion of the falsity of a 
proposition means its simple deletion. The contrariety is not 
between knowledge and knowledge, but between knowledge 
and ignorance. 

It ought also to be remembered, in dealing with the doc 
trine of falsity, that the falsity of all Xs are Ys only 
implies that one or more Xs are not Ys. Now in practice 
one or a few exceptions are often of no importance ; there 
are in many cases singular exceptions which in a sense agree 
with, and in a sense falsify, a general proposition. Thus 
all points of a revolving sphere describe circles, excepting 



xii.] QUESTIONS AND ANSWERS. 119 

the two points at the poles. Other examples of singular 
exceptions will be found in the Principles of Science, Chapter 
XXIX. Professor Henrici points out (Elementary Geometry, 
1879, p. 37) that a proposition must be considered to be 
true in general, if it be true in an infinite proportion of 
cases, and false only in a finite number of exceptions. 

This subject of the truth and falsity of propositions as 
premises and conclusion may be pursued in Karslake s 
Logic, Vol. I. p. 83 ; Whately, Book II. Chap. iii. 2 ; 
Aristotle, Prior Analytics, Book II. Chaps, i. iv. ; Port 
Royal Logic, Part II. Chap. vii. Watts Logic, Part II. 
Chap. ii. 7 and 8. 

Most of the scholastic logicians, such as Thomas Aquinas 
and Nicephorus Blemmidas, treat this subject elaborately. 



9. c Trust (said Lord Mansfield to Sir A. 
Campbell) to your own good sense in form 
ing your opinions ; but beware of attempting 
to state the grounds of your judgments. 
The judgment will probably be right ; the 
argument will infallibly be wrong. 

Explain this phenomenon, and show its logical 
significance. [P.] 

If you give reasons for a decision, implying that those 
reasons are sufficient, and are the reasons upon which you 
did make the decision, it is possible for critics subse 
quently to inquire whether such reasons logically support 
the conclusion derived from them. If they do not, the 
judge will be detected in a paralogism which there may be 
no means of explaining away. But, if no reasons be given, 



120 TRUTH AND FALSITY. [CHAP. 

it will seldom be possible for critics to make any such 
detection. It is impossible, as a general rule, to publish in 
detail the law as well as the evidence upon which a law 
case is decided, and, even if it were published, it would 
generally be impossible to detect bad logic in a man who 
does not assign the precise points on which he relies, and 
the way in which he argues about a complex mass of 
details. 

Although it may be, from his own point of view, con 
venient and discreet for a man to avoid giving reasons for 
any important public decision, if he can avoid it, yet it is 
an open question how far such means of escaping criticism 
is likely to increase the carefulness and impartiality of his 
judgments. There are many cases, including nearly all the 
verdicts given by juries on points of fact, where it would be 
highly undesirable to require any statement of reasons. 
Where the result depends upon oral testimony, the be 
haviour of witnesses, the estimation of degrees of prob 
ability and degrees of guilt, it is quite impossible to define 
and publish the real premises of the conclusion come to. 
We must trust to common sense and judicial tact. The 
same remarks may apply to various arbitrations, magisterial 
decisions, administrative acts, votes of members of de 
liberative bodies. But where the grounds of decision are 
precise and brief, so as to be capable of complete state 
ment, it seems absurd to suppose that a judge will judge 
less well because he needs to disclose his argument. If he 
displays bad logic, where bad logic can be judged, he is 
clearly not fit to be a judge. Lord Mansfield s advice may 
possibly have been prudent and good when given to a man 
who was forced to act in novel circumstances, and in a 
distant colony (Jamaica), where his decisions would have 
more of the nature of administrative acts than law-building 



xii.] QUESTIONS AND ANSWERS. 121 

judgments. But the decisions of the High Court of 
Justice in England not only affect the parties in the cause, 
but shape the public law of a large part of the civilised 
world, and it is of course requisite that they should be 
guided by good logic. 



CHAPTER XIII. 

EXERCISES REGARDING FORMAL AND MATERIAL TRUTH 
AND FALSITY. 

1. COMPARE the following syllogisms, or pseudo- syllo 
gisms, both as regards their formal correctness, and as 
regards the material truth of their premises and conclusion ; 
then explain how it is that a materially true conclusion is 
obtained in each case. 

(1) All existing things are real things ; 
No abstract ideas are existing things ; 

.-. No abstract ideas are real things. 

(2) No real things exist ; 

All abstract ideas are real things ; 
.-. No abstract ideas are real things. 

(3) All real things are existing things ; 
No abstract ideas are existing things ; 

.-. No abstract ideas are real things. 

2. If there be two syllogisms, of which we know that 
their major premises are subcontrary propositions, how 
may we determine the figure and mood of both ? May 
their conclusions be both true in matter ? 

3. Prove by means of the syllogistic rules, that given the 



CHAP, xiii.] EXERCISES. 123 

truth of one premise and the conclusion of a valid syllo 
gism, the knowledge thus in our possession is in no case 
sufficient to prove the truth of the other premise. [c] 

4. It is known concerning a supposed syllogism that it 
involves a fallacy of undistributed middle, and that one of 
the premises is false in matter; can we or can we not draw 
any conclusion under these circumstances ? 

5. Construct two syllogisms, such that the major premise 
of one shall be the subcontrary of the conclusion of the 
other, and such also that the conclusions of both shall be 
true in matter. Are these data sufficient to determine the 
figure ? 

6. If one premise be false in matter, and the syllogism 
correct in form, does it follow that the conclusion is false in 
matter ? 

7. Examine the doctrine that, if the conclusion of a 
syllogism be true, the premises may be either true or false ; 
but that, if the conclusion be false, one or both of the 
premises must be false. 

8. Interpret the logical force of the following passage 
from Mr. Freeman s Essay on the Holy Roman Empire : 
It may have been foolish to believe that the German King 
was necessarily Roman Emperor, and that the Roman 
Emperor was necessarily Lord of the world. 

9. Taking a syllogism of the third figure, and assuming 
one of the premises to be false, show whether or not, with 
the knowledge of its falsehood thus supposed to be in our 
possession, we can frame a new syllogism : if so, point out 
the figure and mood to which it will belong. 

10. What do you mean by (i) Formal, (2) Material 



124 TRUTH AND FALSITY. [CHAP. 

truth, as applying (a) to a single proposition, (b) to a 
syllogism ? 

11. Give a careful answer to the miscellaneous example, 
No. 88, in Elementary Lessons in Logic, p. 322. 

1 2. Is the following extract sense or nonsense, logically 
correct or incorrect ? We may doubt whether the ancient 
method of reduction can prove the validity of any syllogistic 
mood ; for, as from false premises we can illogically obtain 
a true conclusion, the reductio ad impossibile has doubts cast 
upon its validity as a method of proof. 

13. What is the precise meaning of the assertion that it 
is false to say that Castro cannot be proved .not to be 
Orton ? 

14. If P asserts that oxygen, hydrogen, and nitrogen 
cannot be liquefied, and Q denies the assertion, what 
precisely must Q be understood to mean ? 

15. Analyse all that is implied in the assertion of the 
falsity of each of the following propositions : 

(1) Roger Bacon was a giant. 

(2) Descartes died before Newton was born. 

(3) Bare assertion is not necessarily the naked truth. 

(4) All kinds of grass except one or two species are not 

poisonous. 

1 6. Let X, Y, Z, P, Q, R, be six propositions : given 

(a) Of X, Y, Z, one and only one is true ; 

(b) Of P, Q, R, one and only one is true ; 

(c) If X is true, P is" true ; 

(d) If Y is true, Q is true ; 

(e) If Z is true, R is true ; 



EXERCISES. 125 

Prove syllogistically, that 

(/) If X is false, P is false ; 
(g) If Y is false, Q is false ; 
(ti) If Z is false, R is false. [c.] 

17. How do you meet the following difficulties ? 

(1) True premises may by false reasoning give a correct 
conclusion ; because, in a train of reasoning there may be 
two errors, and one error may neutralise the other. 

(2) Since truth applies only to propositions, and a term 
per se is incapable of truth, it follows that a term mustier se 
always be false, because everything must be either true or 
false. 



CHAPTER XIV. 

PROPOSITIONS AND SYLLOGISMS IN INTENSION. 

i. To any one desirous of acquiring a thorough command 
of logical science, nothing is so important as a careful study 
of the intensive or comprehensive meaning of terms, pro 
positions, and syllogisms. This indeed is not an easy task, 
as is shown by the fact that some great logicians who have 
written upon the subject, especially Sir W. Hamilton, have 
fallen into grave errors, or at the best fatal ambiguities of 
expression. Most of the common text-books, again, either 
ignore the subject altogether, or else treat it in a manner 
quite disproportioned to its difficulty and importance. The 
following questions and answers touch some of the more 
obscure points of the matter, but the student is assumed to 
have read the fifth of the Elementary Lessons in Logic, or 
else to have studied the subject in one or more of the 
following books : Port Royal Logic, Part L, Chapters v. to 
vii., Spencer Baynes Translation, 1861, pp. 45 55 (this 
work was the first to draw attention to the subject in modern 
times); Watts Logic, Part I. Chapter vi., 9 and 10; 
Levi Hedge, articles 34 to 38 ; Thomson s Outline of the 
Necessary Laws of Thought, 52; Spalding, 1857, 30 
to 33 ; Walker s Commentary on Murray s Compendium, 
Chapter II. ; Bo wen s Treatise on Logic, Chapter IV. 

The most elaborate treatment of the subject is found in 



CHAP, xiv.] QUESTIONS AND ANSWERS. 127 

Peirce, Proceedings of the American Academy of Aits and 
Sciences, 1867, Vol. VII. , pp. 416 432. If the student 
read Hamilton s Lectures on Logic, he must carefully observe 
what is said about Hamilton in this chapter. 



2. State the proposition Men are mortals in 
the intensive form. 

This proposition, as it stands, is clearly extensive, and 
asserts that all individual men will be found among the 
things called mortals. When asked to turn such a proposi 
tion into the intensive form, students make all kinds of 
blunders, saying, for instance, that All the qualities con 
noted by the term man are connoted by the term mortal, 
or All the properties of men are properties of mortal. 
This is certainly not the case, because men, in addition to 
being mortal, are rational, are vertebrate, are erect, etc. 
Again, a student would say, The attributes of man connote 
the attributes of mortality. This means nothing, the verb 
connote being wrongly used. To say, again, that All 
which possess the properties of man, possess the properties 
of mortal, is to leave the proposition just as it was before, 
All which possess the properties of man/ being simply 
men. 

In passing from the extensive to the intensive mode of 
thought, there must be a complete inversion of the relation 
of the terms. As men are a part of mortals, so the qualities 
of mortals are a part of the qualities of men. If we like 
we may use a different kind of copula ; but, in that case, 
much care is necessary to avoid error. The following are 
different modes of expressing correctly the same truth, the 
first of each pair being an extensive and the second the 
corresponding intensive form of assertion. 



128 INTENSION. [CHAP. 

c All men are included among mortals ; 
-| All qualities of mortals are included among the qualities 
v. of men. 

J Mortals include men ; 
(. Properties of men include the properties of mortals. 

J Man is a species of mortal ; 

( The genus mortal is in the species man. 

( Men are part of mortals ; 

( Mortality is part of humanity. 

3. Can we exhibit particular and negative propo 
sitions in the intensive form ? 

This question has not, I think, been much investigated by 
logicians, and the remarks to be found in the works of 
Hamilton and most other logicians apply only to the 
universal affirmative proposition. Taking the particular 
affirmative, Some crystals are opaque, it asserts that * One 
or more crystals are among opaque things. It follows, no 
doubt, that the quality opaqueness is among the qualities 
of one or more crystals, namely, the particular crystals 
referred to in the extensive proposition. Thus I may be 
treated intensively much as A is treated. 

Taking the negative proposition, * No iron bars are trans 
parent, we cannot infer that No properties of transparent 
objects are properties of iron bars. This inference would 
be quite false ; for, there may be many properties, such as 
gravity, inertia, indestructibility, extension, etc., which are 
possessed alike by transparent objects and iron bars. All 
we can infer is that Not all the properties of transparent 
things are in iron bars, or, Some of the properties of 
transparent things are not in iron bars. Entire separation 
in extension involves only partial separation in intension, or 



xiv.] QUESTIONS AND ANSWERS. 129 

an extensive assertion in E gives an intensive assertion in 
O. We may also change E into A, getting All iron bars 
are non-transparent objects, which of course gives the 
intensive form, * The quality of non-transparency is among 
the qualities of iron bars. 

We may in a somewhat similar way treat the particular 
negative, say Some crystals are not symmetrical. We 
cannot infer that All the common properties of symmetrical 
things are absent from some crystals, but only some of those 
properties. 

4. How shall we state the following syllogism in 
the intensive form ? 

All crystals are solids ; 
All topazes are crystals ; 
Therefore, All topazes are solids. 

We must transmute each of the three propositions into 
the intensive form, and transpose the premises, thus 

All qualities of crystals are qualities of topazes : 
All qualities of solids are qualities of crystals ; 
Therefore, All qualities of solids are qualities of topazes. 

The syllogism is turned, as it were, completely inside 
out. 

5. Is Hamilton correct in stating the following 
as a valid syllogism ? 

S comprehends M; 
M does not comprehend P ; 
Therefore, does not comprehend P. 

Professor Francis Bowen (Logic, p. 237) has pointed out 
that, as the statement stands (see Hamilton s Lectures, Vol. 

K 



130 INTENSION. [CHAP. 

III. pp. 315, 316, American Edition, p. 223), it is simply 
illogical. It involves a fallacy of Illicit Process of the 
Major Term ; in short, S may comprehend P through 
others means besides M. But Professor Bowen thinks that 
Hamilton s error lay in the choice of language, which was 
such as no one would understand it as Hamilton really in 
tended it. The matter, however, is too important to be 
passed over in this way, and I proceed to notice other 
places in which Hamilton has treated of intensive 
syllogisms. 

In his sixteenth Lecture on Logic (Vol. III. p. 295), we 
find the following : 

An Extensive Syllogism. An Intensive Syllogism. 
B is A C is B 

C\$ is A 

C is A C is A 

All man is mortal ; Caius is a man ; 

But Caius is a man ; But all man is mortal ; 

Therefore, Caius is mortal. Therefore, Caius is mortal. 

Between the syllogisms as thus stated there is no difference 
whatever, except the transposition of premises, a mere 
difference in the order of writing which is immaterial to the 
point in question. It is true that Hamilton goes on to 
explain as follows : 

< In these examples, you are aware, from what has pre 
viously been said, that the copula in the two different 
quantities is precisely of a counter meaning ; in the quantity 
of extension, signifying contained under ; in the quantity of 
comprehension, signifying contains in it 

Afterwards the example of a concrete syllogism before 
given, is thus fully stated in the intensive form (p. 296). 



xiv.] QUESTIONS AND ANSWERS. 131 

* The Major term Caius contains in it the Middle term 

man ; 
But the Middle term man contains in it the Minor term 

mortal ; 
Therefore, the Major term Caius contains in it the Minor 

term mortal. 

To say the least, this is a very clumsy and misleading 
mode of explanation ; for after all, the point of the matter 
is left untouched, namely, that it is individual things which 
are contained under in the extensive sense, and qualities, or 
attributes, which are contained in the other sense. Is it 
not absurd to say that Caius contains man, without ex 
plaining that man is here taken intensively ? A thing does 
not contain any of its qualities in the same way that the 
class man, extensively regarded, contains one of its members 
or significates, namely, Caius. Nor is the matter much 
mended by referring back to the previous explanation 
(p. 274), where Hamilton illustrates the relations, saying, 
Thus the proposition, God is merciful, viewed in the one 
quantity, signifies God is contained under merciful, that is, the 
notion God is contained under the notion merciful ; viewed 
as in the other, means, God comprehends merciful, that is, 
the notion God comprehends in it the notion merciful? This 
is again all wrong, unless we interpret notion and containing 
in a totally different sense in the second as compared with 
the first statement. Even if Hamilton understood the 
matter correctly himself, he ought to have stated it unequi 
vocally, and not to have left the reader to put the matter 
right by careful interpretation of the same words in two 
different senses. His subsequent exposition of the sorites 
is even worse ; for he gives the identically same premises 
twice over in different order, and asserts, without other 
explanation, that one is in comprehension and the other in 



132 INTENSION. [CHAP. 

extension. Suppose it were stated in a police court that 
Brown struck Robinson, and then Robinson struck 
Brown. Should we not be surprised to learn subsequently 
that the verb struck was used in a psychological sense in 
one case and a physical one in the other, the real meaning 
being that Brown struck Robinson as a very disagreeable 
fellow, and then Robinson struck Brown on the head ? 
Yet this would not be worse than for Hamilton to state a 
syllogism or sorites twice over, with an unimportant change 
of order, and then assume that the reader takes one state 
ment to be in extension and the other in intension. 

I am obliged, therefore, to coincide with the opinion of 
De Morgan that Hamilton really missed the point of the 
question, in short, did not understand what he was writing 
about. The whole matter is put in the clearest light by the 
following few lines from De Morgan s Syllabus, pp. 62-3 : 

The logicians who have recently introduced the distinc 
tion of extension and comprehension, have altogether missed 
this opposition of the quantities, and have imagined that 
the quantities remain the same. Thus, according to Sir 
W. Hamilton, "All X is some Y" is a proposition of com 
prehension, but "Some Y is all X" is a proposition of 
extension. In this the logicians have abandoned both 
Aristotle and the Laws of Thought from which he drew the 
few clear words of his dictum : " the genus is said to be part 
of the species ; but in another point of view (aAAws) the 
species is part of the genus." All animal is in man, notion 
in notion : all man is in animal, class in class. In the first, 
all the notion animal part of the notion man ; in the second, 
all the class man part of the class animal. Here is the 
opposition of the quantities. 

The same view is more fully stated in De Morgan s Third 
Memoir on the Syllogism (pp. 17-19; Camb. Phil. Trans. 



xiv.J QUESTIONS AND ANSWERS. 133 

1858, pp. 188-9). On the whole, I conclude that Hamil 
ton s treatment of the subject is so doubtful and confusing 
that it had better not be studied in an elementary course of 
reading. 

De Morgan, in the paper just referred to, gives some 
remarks about the history of the doctrine of intension and 
extension, and speaks of Hamilton as a logician who has 
recently contended for the revival, or rather the full intro 
duction, of the distinction of extension and comprehension. 
He correctly names the Port Royal Logic as being the first 
modern work to insist on the distinction, though the use 
made of it is * not very extensive. But he names only one 
other work, the Institutiones Philosophies of J. Bouvier 
(.3d Ed. Mans, 1830), as describing this distinction. 
De Morgan s reading of modern logic was not extensive. 
Not to mention the familiar Watts Logic, in which the 
doctrine is frequently dwelt upon (see Part I. Chap. III. 
Section 3 ; Chap VI. Section i o, and elsewhere), I 
find the matter excellently explained in 1816 in the 
brief manual of the American logician, Levi Hedge (pp. 
42-44). In Murray s Manual, formerly much used in 
Dublin and Glasgow, the subject is fully explained, and 
in the clearest possible manner, in the Commentary of John 
Walker on Chapter II. This is in fact one of the best 
pieces of logical exposition which I know. Walker re 
marked that he had treated the point fully, because he 
regarded it as absolutely necessary to the understanding of 
the subsequent pages, which were often puzzling to students 
not familiar with the distinction between the comprehension 
and extension of a term. With some regret I must hold, 
then, that the pretensions of Hamilton in this matter are 
mistaken and unfounded. 

The whole subject of extension and comprehension or 



134 INTENSION. [CHAP. xiv. 

intension has been investigated with much care and pro 
fundity of thought by the American logician, Professor 
C. S. Peirce, in the memoir already referred to (see p. 127). 
This memoir should be studied by those who wish to 
acquire a thorough understanding of logical principles and 
relations. 

6. It is asserted by some logicians that the 
predicate of a proposition must be interpreted 
in intension while the subject is regarded in 
extension. Give your opinion upon this point, 
and explain the bearing of the question upon 
recent logical controversies. [c.] 

I should answer this question to the effect that a propo 
sition being, conformably to the opinion of Condillac, 
necessarily of the nature of an equation, it is absurd to 
suppose that things can be equated to their own qualities 
or circumstances. A proposition in extension expresses the 
identity of a thing or class of things with the same thing 
or class under another designation. As De Tracy says 
(Id eologie, Vol. III. p. 529), Dans tout jugement, les deux 
idees comparees sont necessairement egales en extension? 
A proposition in intension expresses an identity between 
the attributes of the one member and those of the other. 
The subject may be pursued in my Essay on Pure Logic, 
or the Logic of Quality apart from Quantity, 1864, passim ; 
in J. S. Mill s Logic, Book I. Chapter V. ; and in Dr. 
Martineau s review of Samuel Bailey on the Theory of 
Reasoning, in his Essays Philosophical and Theological, 1869, 
Vol. II. 



CHAPTER XV. 

QUESTIONS ON INTENSION. 

T. Christian/ animal, Episcopalian, organised, 
man. Arrange these terms (i) in the order of exten 
sion, beginning with the most extensive; and (2) in the 
order of comprehension, beginning with the most com 
prehensive. [L.] 

2. Arrange the following in the same manner : General, 
animal, composer for the pianoforte, Roman, historian of 
his own campaigns, conqueror of Gaul. (See De Morgan, 
Third Memoir, pp. 20, 21.) 

3. Arrange in order of extension and intension such of 
the terms given in Question i of Chapter II. as are the 
names of subaltern, genera, and species, and can be arranged 
in a series. 

4. Analyse the following terms in the counter quantities 
or wholes of extension and intension : Man, government, 
law, triangle, vegetable. [L.] 

5. Show that the analysis of an intensive equals the 
synthesis of an extensive whole. [c.] 

6. Invent a syllogism in Barbara, and state it both in the 
extensive and in the intensive forms. [L.] 



136 INTENSION. [CHAP. xv. 

7. What is the place of the Major and Minor Terms in 
the conclusion of (a) an extensive, and (b) an intensive 
(comprehensive) syllogism ? 

8. Can the distinction of extension and intension be 
made to apply to the inductive syllogism? [c] 

9. Select from pp. 91 to 98 examples of the moods 
Celarent, Cesare, and Camenes, and state them in the 
intensive form. 

10. What is the difference of meaning of genus and 
species in extent and intent ? Is the extent of a notion 
always less as the intent is greater, and vice versa ? 

1 1. Interpret the following propositions in extension and 
intension : 

A libel is a malicious and injurious statement. 

He who believes himself to be always right in his opinion 

claims infallibility. 

It is impossible to be and not to be. 
He that can swim needs not despair to fly. 



CHAPTER XVI. 

HYPOTHETICAL, DILEMMATIC, AND OTHER KINDS OF 
ARGUMENTS. 

i. SOME attempt will be made in the subsequent chapters 
on the Elements of Equational Logic to illustrate the 
actual and possible variety of assertions and arguments. 
But it will be convenient to give here a few examples of 
hypothetical and other arguments in the less common forms. 
Several subtle questions arising out of the hypothetical 
form of assertion are also considered with some care ; 
but it has not been thought necessary to treat all the 
various forms of disjunctive and dilemmatic arguments 
which will be found described in almost identical terms 
in numerous text-books. 

2. If virtue is voluntary, vice is voluntary ; but 
virtue is voluntary ; therefore so is vice, [w.] 

A valid Constructive Hypothetical syllogism, equivalent 
to the following categorical one in Barbara : 

Beings who can be virtuous at will can also be vicious 

at will ; 

Men can be virtuous at will ; 
Therefore, they can be vicious at will. 



138 HYPOTHETICAL ARGUMENTS, ETC. [CHAP. 

3. Logic is indeed worthy of being cultivated, 
if Aristotle is to be regarded as infallible ; but 
he is not : Logic, therefore, is not worthy ot 
being cultivated. [w.] 

Clearly a false hypothetical syllogism. The antecedent 
is, if Aristotle is to be regarded as infallible ; this is denied 
in the minor premise. In the categorical form the pseudo- 
argument might be stated somewhat as follows : 

Those who regard Aristotle as infallible must consider 

logic worthy of being cultivated ; 
We do not regard Aristotle as infallible ; 
Therefore, we do not consider logic, etc. 

There is Illicit Process of the Major Term. 

4. We are bound to set apart one day in seven 
for religious duties, if the fourth command 
ment is obligatory on us : but we are bound 
to set apart one day in seven for religious 
duties ; and hence it appears that the fourth 
commandment is obligatory on us. [w.] 

The antecedent is * if the fourth commandment is obli 
gatory ; the consequent is we are bound, etc. ; it is the 
consequent which is affirmed, so that the argument involves 
the Fallacy of Affirming the Consequent. It may be put 
categorically as follows : Those on whom the fourth com 
mandment is obligatory are bound, etc. ; we are bound, etc. ; 
therefore, we are among those on whom the fourth com 
mandment is obligatory. The fallacy is evidently that of 
Undistributed Middle, the pseudo-mood being A A A in 
the second figure. 



xvi.] QUESTIONS AND ANSWERS. 139 

5. (i) If the prophecies of the Old Testament 
had been written without knowledge of the 
events of the time of Christ, (2) they could 
not correspond with them exactly ; (3) and if 
they had been forged by Christians, (4) the} 
would not be preserved and acknowledged by 
the Jews : (5) they are preserved and acknow 
ledged by the Jews, (6) and they correspond 
exactly with the events of the time of Christ : 
therefore they were (7) neither written without 
knowledge of those events, (8) nor were forged 
by Christians. [w.] 

The above argument will be found to consist of two 
valid destructive hypothetical syllogisms woven together in 
statement. Thus (i) and (2) are the antecedent and 
consequent of the first syllogism ; (6) is its negative 
minor, and (7) is its negative conclusion. The second 
syllogism has (3) and (4) for its antecedent and consequent, 
(5) for its negative minor premise, and (8) for its con 
clusion. 

6. In how many ways can you state the sub 
stance of the categorical proposition A wolf 
let into the sheep-fold will devour the sheep ? 

Isaac Watts, in his Essay on the Improvement of the 
Mind, has well pointed out the variety of expression which 
may be given to the same real assertion. Thus, as equiva 
lents for the above proposition, he gives the following : 
If you let a wolf into the fold, the sheep will be de 
voured : The wolf will devour the sheep, if the sheep-fold 
l)e left open : If the fold be not left shut carefully, the 



i 4 o HYPOTHETICAL ARGUMENTS, ETC. [CHAP. 

wolf will devour the sheep : The sheep will be devoured 
by the wolf, if it find the way into the fold open : There 
is no defence of the sheep from the wolf, unless it be kept 
out of the fold : A slaughter will be made among the 
sheep, if the wolf can get into the fold. There are various 
modes of hypothetically stating the result contained in the 
categorical original. 

7. In a strictly logical point of view, ought it to 
be offensive to Captain Jones to say of him 
If Captain Jones does run away in battle, he 
will live to fight another day ? 

This question touches deeply, not only the soldierly repu 
tation of Captain Jones, but, what is much more important, 
the precise import of propositions. It puts forward Captain 
Jones as running away in battle, but it puts this forward 
only as a hypothesis, the result of which would be his living 
to fight another day. It is quite a different matter, what 
meaning such a proposition might be taken to imply in 
common life ; things are often said in the form of innuendo. 
The mere coupling of a man s name with a disreputable 
action, even though the action were expressly denied of him, 
raises the question, Why was the assertion made at all 
unless to bring the terms together in the mind of the hearer? 
If in company a gentleman were suddenly to remark There 
is not the least reason to believe that Captain Jones did run 
away in his last action ; here is a point blank denial of 
any ground for believing an assertion to that effect ; yet 
every one would construe such a mal-a-propos denial as 
evidence of a wish to raise the question, and possibly start 
a rumour, which would presently take a disagreeable affirma 
tive form. Thus we see that the logic of conversation is 



xvi.] QUESTIONS AND ANSWERS. 141 

widely different in apparent nature from the strict logic of 
science ; not that it is really different in the end, when 
thoroughly analysed. But we constantly deal with illogical, 
inaccurate, or even untrustworthy persons, so that we can 
seldom be sure that an assertion will be construed and 
repeated in the form which we originally gave to it. There 
is too much truth in the saying of Talleyrand, that words 
were given to us to disguise our thoughts. 

8. If Brown says to Jones, Because Robinson is 
foolish you have no need to be foolish, does 
Brown assert categorically that Robinson is 
foolish ? 

There can be no doubt that, in the logic of common life, 
Brown would be understood to make an imputation upon 
the wisdom of Robinson, especially if the remark was not 
explained by the previous course of the conversation. But 
in strict logic it seems very doubtful whether the conjunction 
because should be interpreted differently from if, as in 
the last question. The fact of Robinson being foolish is 
no reason, etc. Foolishness on the part of Robinson is 
no reason for you being foolish. A logical copula must 
not be understood to assert the physical existence and 
occurrence of its subject or predicate; it only asserts a 
relation between them. 

9- If P is 2, and Q is R, it follows that P is R : 
but suppose it to be discovered that no such 
thing as Q exists, how is the truth of the 
conclusion, P is R, affected by this discovery ? 

I do not see how there is in deductive logic any question 
about existence. The inference is to the effect that if the 



142 HYPOTHETICAL ARGUMENTS, ETC. [CHAP. 

propositions P is Q and Q is R are true, then the conclusion 
P is R is true. The non-existence of Q may possibly 
render one or both premises materially false, in which case 
the reasoning vanishes, but is not logically defective. If 
I argue, for instance, that satyrs are creatures half man and 
half goat ; and creatures half man and half goat are very 
hideous, therefore satyrs are very hideous ; the reasoning is 
equally good whether satyrs exist or not. We cannot, of 
course, say that the conclusion is materally true, if there 
be no objects to which the material truth can apply. But 
if I argue that satyrs are creatures half man and half goat, 
and such creatures exist in Thessaly, therefore, satyrs exist 
in Thessaly ; in this case the non-existence of the middle 
term would affect the material truth of the second premise, 
and, if this be held false, we cannot affirm the material 
truth of the conclusion. 

I ought to add that De Morgan in more then one place 
assumes that the middle term must have existence, or even 
objective existence; thus he says (Syllabus, p. 67): In all 
syllogisms the existence of the middle term is a datum] etc. 
This is one of the few points in which it is possible to 
suspect him of unsoundness. 

The student may refer to Hamilton s Lectures, Vol. III. 
pp. 454-5, and p. 459, on Sophisms of Unreal Middle; 
see also Whately s Analytical Outline, 3. 

10. Lias lies above red sandstone; red sand 
stone lies above coal ; therefore lias lies above 
coal. [W.] 

This is one of many examples to be found in the logic 
books of arguments which simulate the syllogistic form. 
It is often said that they can be solved syllogistically ; but 



QUESTIONS AND ANSWERS. 143 

certainly this cannot be done by the ordinary rules and 
processes of the syllogism. The most that we can get, 
even by substitution, is that Lias lies above what lies above 
coal. The fact is that the argument is really a mathematical 
one, involving simple equations. It is precisely similar to 
one which has been thus treated by Professor F. W. Newman 
(Miscellanies, 1869, p. 28), and Mr. J. J. Murphy. The 
former of these logicians, as quoted by the latter, remarks 
* The argument Lead is heavier than silver ; Gold is heavier 
than Lead : therefore Gold is heavier than Silver^ brings to 
the mind conviction as direct as the simplest of syllogisms. 
To say that its validity depends on its being reducible to 
syllogism, is wholly unplausible : for to effect the reduction, 
you have to make changes of form at least as hard to 
accept as the direct argument : and when you have got your 
syllogisms, they are more complicated and cumbrous than 
the argument as it stands. 

Mr. Murphy (The Relation of Logic to Language : Belfast 
Natural History and Philosophical Society, i yth February 
1875) treats the argument simply as a question of quantity, 
thus 

Call the weights of gold, lead, and silver respectively 
x, y, and z : then x = y + / 

y = * + q 
x = z + q + /. 

In the old logic, the foregoing conclusion could be drawn 
only by means of the following syllogism : 

That which is greater than the greater is greater than the 
less : 

The weight of gold is greater than that of lead, and the 
weight of lead greater than that of silver : 

1 Newman has inadvertently written Lead is heavier than Gold, which 
is wrong as to fact. 



144 HYPOTHETICAL ARGUMENTS, ETC. [CHAP. xvi. 

Therefore the weight of gold is greater than that of 
silver. 

Considered as fact all this of course is true, but considered 
as logic it is wrongly stated. That which is here stated as 
the major premise is really the syllogistic canon. It is not 
merely a general truth, like the truth that all matter gravi 
tates, but a logical principle, lying as near to the first 
principles of the science as the axiom that a part of a part 
is a part of the whole. 

We have only to assume x to be the height of lias, y the 
height of red sandstone, and z of coal above any one fixed 
datum line, and the same equations represent the argument 
at the head of this section. 

It may be added that Reid was doubtless right in denying 
that we argue syllogistically when we infer that because A 
and C are both equal to B they are equal to each other. 
We may throw it into the form, Things equal to the same are 
equal to each other ; A and C are things equal to the same, 
therefore they are equal to each other. But this is a delu 
sive syllogism. The inference is really accomplished in 
obtaining the major premise. The inferences of equality 
are prior to and simpler than the inferences of logic, and 
the attempt of Herlinus and Dasypodius to throw Euclid 
into the syllogistic form has been rightly ridiculed, because 
it is an attempt to prove the more simple and self-evident 
by means of the more complex. 

Some remarks on this point will be found in De Morgan s 
Second Memoir on the Syllogism, 1850, pp. 50, 51; his 
Fourth Memoir, 1860, p. 8, etc.; in Mr. Murphy s paper 
quoted above ; and in Hallam s remarkable note to Section 
129 of Vol. III., Chapter III., of his Introduction to the 
Literature of Europe (ist Ed. p. 288 ; 5th Ed. p. in). 



CHAPTER XVII. 

EXERCISES IN HYPOTHETICAL ARGUMENTS. 

i. (i) IF he is well, he will come : he is not well : there 
fore he will not come. 

(2) If he is well, he will come : he will come : therefore 

he is well. [H.] 

(3) I am sure he will not come, for he is not well ; and if 

well he would come. 

(4) He will write if he is well ; but as he is not well, 

therefore he will not write. 

Analyse the above arguments and point out which are 
fallacious, and why. 

2. Into how many forms of expression can you throw the 
matter of this proposition ? Sulphuric acid combined with 
calcium produces gypsum. 

3. Throw into the form of hypothetical propositions the 
following disjunctives 

(1) Either the Claimant is Ortoh, or many witnesses are 

mistaken. 

(2) The tooth of a mammalian is either an incisor, canine, 

bicuspid, or molar tooth. 
L 



146 HYPOTHETICAL ARGUMENTS. [CHAP. 

4. Under which of the commonly recognised forms of 
syllogism would you bring the following ? 

If A is-ff, Cis>; 

If Cis >, is F ; 
Therefore, If A is B, E is F. [c.] 

5. Are hypothetical propositions capable of conversion ? 
If so, convert these 

(1) If it has thundered it has lightened. 

(2) Unless it has lightened it has not thundered. 

6. Which of the following arguments are logically 
correct ? 

(1) A is , if it is C ; it is not C, therefore it is not >. 

(2) A is not B unless it is C ; as it is not C, it is not B. 

(3) If A is not B, C is not D; but as A is B, it follows 

that C is D. 

(4) A is not B, if C is D / C then is not D, for A is B. 

7. If the Hypothetical Modus Ponens and Modus Tollens 
are taken as corresponding to the Categorical First and 
Second Figures, and their typical forms to the Moods 
Barbara and Camestres, respectively, what other forms of 
the respective Hypothetical Modi would correspond to the 
other moods of the respective Categorical Figures ? [R.] 

8. If A is true, B is true ; if B is true, C is true ; it C is 
true, D is true. What is the effect upon the other assertions 
of supposing successively that (i) D is false; (2) that C is 
false ; (3) that B is false ; (4) that A is false ? 

9. Analyse the following arguments and estimate their 
validity. 



xvii.] EXERCISES. 147 

(1) I shall see you if you do not go; but as you are 

going I shall not. 

(2) The Penge convicts were guilty of murder, if, after 

long continued neglect at their hands, Harriet 
Staunton died. 

(3) Since the virtuous alone are happy, he must be 

virtuous if he is happy, and he must be happy if 
he is virtuous. 

(4) If there were no dew the weather would be foul : but 

there is dew ; therefore the weather will be fine. 

[a] 

(5) If there are sharpers in the company we ought not to 

gamble ; but there are no sharpers in the com 
pany ; therefore we ought to gamble. [E.] 

(6) I could then only be accused with justice of acting 

contrary to my law, if I maintained that Mursena 
purchased the votes, and was justified in doing so. 
But I maintain that he did not buy the votes, 
therefore, I do nothing contrary to the law. 
Cicero, Pro L. Mursena, c. iii. (See Devey s Logic, 
1854, p. 133.) 

10. State in the form of a disjunctive argument the matter 
of the First Book of Samuel, chapter xvi. verses 6-13. 

1 1. Examine the question whether hypothetical and dis 
junctive arguments are reducible to the forms of the 
categorical syllogism. 

12. Dilemmatic arguments are more often fallacious than 
not. Why is this ? [c.] 

13. Investigate the logical position of the parties to the 
following colloquy from Clarissa Harlowe : Morden 
But if you have the value for my cousin that you say you 



148 HYPOTHETICAL ARGUMENTS. [CHAP. xvn. 

have, you must needs think . Lovelace You must allow 

me, sir, to interrupt you. If I have the value I say I have ! 
I hope, sir, when I say I have that value, there is no cause 
for that if t as you pronounced it with an emphasis. Morden 
Had you heard me out, Mr. Lovelace, you would have 
found that my if was rather an if of inference than of 
doubt. 

This passage is quoted and discussed by Professor Groom 
Robertson in Mind, 1877, Vol. II. pp. 264-6. 



CHAPTER XVIII. 

THE QUANTIFICATION OF THE PREDICATE. 

i. As explained in the preface, I have thought it well to 
discuss and illustrate in this book of exercises, the forms of 
logical expression and inference recognised by Dr. Thomson 
and Sir W. Hamilton. These correspond in most cases 
with what De Morgan represented under different systems of 
notation. They also correspond to some of the expressions 
and arguments current in ordinary life. Although in a scien 
tific point of view it is far better to eliminate the logical will-of- 
the-wisp some, yet the student is obliged to make himself 
acquainted with the pitfalls into which it is likely to lead him. 

It is assumed that the reader has studied the brief 
account of the Quantification of the Predicate given in the 
22nd of the Elementary Lessons in Logic, and he is re 
commended to read, on the same subject, either Thomson s 
Outline, or else Bowen s account of Hamilton s Logic 
(Bowen s Logic, Chapter VIII. ). The study of De Morgan s 
and Hamilton s own writings is a more arduous and 
hazardous undertaking. 

The following are the eight kinds of propositions re 
cognised by Hamilton, as described by Dr. Thomson. 



Sign. Affirmative. 

U All X is all Y. 

I Some X is some Y. 

A All X is some Y. 

Y Some X is all K 



Negative. Sign. 

No X is Y. E 

Some X is not some Y. o> 

No X is some Y. -q 

Some X is no Y. O 



150 QUANTIFIED PREDICATE. [CHAP. 

2. Indicate by the technical symbols the quan 
tity and the quality of the following propo 
sitions : 

(1) All primary forces are attractive. 

(2) All vital actions come under the law 

of habit, and none but vital actions 
do. 

(3) The best part of every man s education 

is that which he gives himself. 

(4) Only ungulate animals have horns. 

(5) Mere readers are very often the most 

idle of human beings. 

(6) Most water -breathing vegetables arc 

flowerless. [p.] 

(1) Is clearly a universal affirmative (A). 

(2) As regards its first part is also A; but the exclusive 

addition, None but vital actions do, means that, 
all not vital actions do not. The two parts 
together yield a proposition in U, all vital actions 
are all that come under the law of habit. 

(3) The best part, being a superlative, is a singular 

term, and so is the predicate that part which, etc. 
Hence the proposition is an identity in U. 

(4) An exclusive proposition equivalent to all not un 

gulate animals have no horns, which is the con- 
trapositive of, and equivalent to, all horned 
animals are ungulate. 

(5) Means that a great many mere readers are, etc., and 

is in the form I. 

(6) Is also a particular affirmative proposition. 



xviii.] QUESTIONS AND ANSWERS. 151 

3. Does not the proposition Y of Thomson imply 
O, that is to say, does not some P is all Q 
imply that some P is not Q ? 

This seems very plausible, because if some P makes up 
the whole of Q, there is, so to say, no room left in <2 s 
sphere for any more Ps, the remainder of which must there 
fore be not Q. This argument, however, overlooks the fact 
that the some P in question may possibly be the whole of 
P, so that there may be no remainder excluded from Q. 

4. Is the proposition Some men are animals 
true ? [E.] 

The proposition is true or untrue materially according to 
the sense we put upon this troublesome word some. If 
we take it to mean one or more it may be all, the pro 
position is true in fact, but of course states less than is 
known to every one. 

We must carefully distinguish between the strict and 
necessary logical interpretation of some, and that which 
applies in colloquy. De Morgan says (Formal Logic, p. 4), 
In common conversation the affirmation of a part is meant 
to imply the denial of the remainder. Thus, by " some of 
the apples are ripe," it is always intended to signify that 
" some are not ripe." There is no difficulty in providing in 
formal logic for this use of the word by stating explicitly 
the two propositions which are colloquially merged into 
one. Thus some of the apples are ripe is really I + O. 

5. What results would follow if we were to in 
terpret * some As are Bs as implying that 
* some other As are not Bs ? 



152 QUANTIFIED PREDICATE. [CHAP. 

The proposition some As are s is in the form I, and 
according to the table of opposition (p. 31) I is true if A is 
true ; but A is the contradictory of O, which would be the 
form of some other As are not .Z?s. Under such cir 
cumstances A could never be true at all, because its truth 
would involve the truth of its own contradictory, which is 
absurd. 

Briefly If A is true, I is true ; and if I implies O, then 
A implies the truth of its own contradictory O. 

Several logicians have come to grief over this troublesome 
word, notably Sir W. Hamilton, who in holding that some 
is formally exclusive of all and none, throws all logical 
systems into confusion. Woolley commits the same great 
mistake in saying (p. 77), In every particular proposition, 
therefore, the affirmative and negative mutually imply each 
other : if only some A is B, then some A is not -B, and 
vice versa? 

6. Explain the precise meaning of the propo 
sition * Some Xs are not some Ys (the 
proposition of Thomson). What is its 
contradictory ? Give your opinion of its 
importance. 

This is one of the eight forms of proposition which 
Hamilton, in pursuance of the thoroughgoing quantification 
of the predicate, introduced into his system. Now, if 
some Y J means any some K, that is to say, if the some 
is undetermined and may be any where in the sphere of Y, 
this proposition does not differ from some Xis not any Y, 
which is the proposition O of the old Aristotelian Logic. 
But if some Y is a determinate part of the class Y, less 
than the whole, then the proposition becomes a mere empty 



xviii.] QUESTIONS AND ANSWERS. 153 

truism ; for, however X and Y may be related, some part of 
X will be different from some part of Y. Thus all equi 
lateral triangles are all equiangular triangles, yet some equi 
lateral triangles are not some equiangular ones. If all John 
Jones sons are Rugby boys, yet some of John Jones sons 
are not some Rugby boys. We see that this proposition w 
is consistent with all the other propositions of the system, 
in all cases, as De Morgan remarks (Syllabus, p. 24), in 
which either X or Y has two or more instances in existence : 
its contrary is " X and Y are singular and identical ; there 
is but one X, there is but one F, and X is Y." A system 
which offers an assertion and denial which cannot be con 
tradicted in the same system carries its own condemnation 
with it, as well observed by De Morgan. 

Archbishop Thomson also rejected this form of propo 
sition. He says : If I define the composition of common 
salt by saying, " common salt is chloride of sodium," I 
cannot prevent another saying that " some common salt is 
not some chloride of sodium," because he may mean that 
the common salt in this salt-cellar is not the chloride of 
sodium in that. A judgment of this kind is spurious upon 
two grounds : it denies nothing, because it does not prevent 
any of the modes of affirmation ; it decides nothing, inas 
much as its truth is presupposed with reference to any pair 
of conceptions whatever. (Outline of the Laws of Thought, 
1860, 79, p. 137.) 

SPALDING, pp. 83, 97-102, etc., symbolises the propo 
sition (o, by \ O. 

In an examination, candidates almost invariably say that 
all Xs are Fs, or all Xs are all Fs, is the contradictory of 
some Xs are not some Fs ; and De Morgan (1863, p. 4) 
speaks of an unnamed logical author who spoiled his work 
with a like blunder. 



154 QUANTIFIED PREDICATE. [CHAP. 

The chief interest of this proposition u> arises from its 
important bearing upon the value of Hamilton s System of 
Logic, and his position as a logician. Hamilton insisted 
upon the thoroughgoing quantification of the predicate, which 
means the recognition and employment of all the eight pro 
positions which the introduction of the quantified predicate 
renders conceivable. Thus was the key-stone to be put 
into the arch of the Aristotelic logic. But if, as Thomson 
and De Morgan seem to me to have conclusively shown, 
this proposition, w, is valueless and absurd, the key-stone 
crumbles and the arch collapses. The same ruin does not 
overtake De Morgan s system, because his eight propositions 
are not all the same as those of Hamilton ; nor does it 
affect in any appreciable degree the views of Thomson and 
George Bentham, who did not insist upon the thoroughgoing 
quantification of the predicate. 

De Morgan has admirably expressed the inherent ambi 
guity of this word. He says (Fifth Memoir on the Syllogism, 
1863, p. 4), " He has got some apples" is very clear: 
ask the meaning of "he has not got some apples," in a 
company of educated men, and the apples will be those of 
discord. Some will think that he may have one apple ; 
some that he has no apple at all; some that he has not got 
some particular apples or species of apples. 

The subject of particular propositions may be pursued in 
Spalding s Logic, 1857, p. 172, and elsewhere; Shedden s 
Logic; Hughling s Logic of Names, 1869, p. 31; Thomson s 
Outline, fifth edition, section 7 7 ; Hamilton s Lectures, vol. 
iv., pp. 254, 279; Devey, 1854, pp. 90-94; De Morgan, 
1863, Fifth Memoir. 

Mr. A. J. Ellis is particularly exact in his treatment of 
this question in his articles in the Educational Times, 
1878. 



xvm.] QUESTIONS AND ANSWERS. 155 

7. Solly says (p. 73) If the premises are "some 
B is A, some C is not B" the reason may 
logically deduce that some C is not some A. 
But this conclusion is not in one of the four 
legitimate forms. Is the argument valid in 
the quantified syllogism, and if so, in what 
mood ? 

The propositions are as follow : 

Some B is some A I 

Some C is not (any B) O 
Some C is not some A w 

The middle term is distributed once in the minor premise, 
and, as both terms of o> are particular, there can be no illicit 
process. One premise is negative, and so is the conclusion. 
No rule of the syllogism is broken, and the argument is 
therefore valid. It appears as I O w in the sixth mood of 
the first figure of Thomson s table. 

8. Which of the following conjunctions of pro 
positions make valid syllogisms ? In the case 
of those which you regard as invalid, give your 
reasons for so treating them. 

First Figure. Second Figure. Third Figure. 

EYO UOto AtoQ 

AEE rj \J O YEO 

I U , [c.] 

The pseudo-mood A E E in the first figure gives illicit 
process of the major term, because the conclusion E distri 
butes its predicate, and the major premise A does not. 
The pseudo-mood I U t\ draws a negative conclusion, rj, 



156 QUANTIFIED PREDICATE. [CHAP. 

from two affirmative premises, but is by oversight given in 
Thomson s Table of Modes, figure i, mode xii., second 
negative form. It is an obvious misprint for I E ^. (Out 
line of the Laws of Thought \ section 103, 5th ed., p. 188.) 
In the table as reprinted in the Elementary Lessons in Logic, 
p. 1 88 (accidentally the same page as in Thomson!), the 
error was corrected in the fifth and later editions. It was 
pointed out to me by Mr. A. J. Ellis. 

E Y O is valid in the first figure. 

In the second figure U O w breaks no rule, but the con 
clusion instead of being w (some Xs are not some Zs), 
might have been in the stronger form O (some Xs are not 
any Zs). The moods U O O and U w w appear in 
Thomson s table, column 4, though U O w does not. The 
mood t] U O is valid. 

In the third figure A w O is subject to illicit process of 
the major term, since the conclusion O distributes its predi 
cate, which is the undistributed predicate of A in the major 
premise. Y E O is not subject to the same objection, 
because Y distributes its predicate ; but, in this last case, 
the conclusion is weakened, and might have been E; hence 
Y E E appears in Thomson s table, tenth mood of third 
figure, and Y E O does not. appear. 

9. In what mood is the following argument : 
Aliquod trilaterum est sequiangulum ; omne 
triangulum est (omne) trilaterum ; ergo, ali- 
quod triangulum est sequiangulum ? 

The first premise, some trilateral figure is an equiangular 
figure, is plainly a proposition in I ; the second, all 
triangles are all trilateral figures, is as plainly a doubly 
universal proposition in U; the conclusion, some triangular 



QUESTIONS AND ANSWERS. 157 

figure is equiangular, is in I. The middle term, trilaterum. 
is distributed in the minor premise, though not in the major; 
there is no illicit process, nor other breach of the syllogistic 
rules, so that the argument is a valid syllogism in the mood 
I U I of the first figure. It appears as the twelfth mood 
in the first column of Thomson s table of moods. See, 
however, Baynes New Analytic, 1850, pp. 126-7, whence 
this example is taken. 

10. Does the following argument fall into any 
valid mood of the syllogism ? 
Some man is all lawyer ; 
Any lawyer is not any stone ; 

therefore, Some man (i.e. lawyer) is not any stone 
(i.e. all the rest are stone). 

This example is taken by De Morgan (1863, p. 10) as a 
case of Hamilton s mood IV. b, as stated in his Lectures on 
Logic, Vol. IV., p. 287, thus, A term parti-totally co- 
inclusive, and a term totally co-exclusive, of a third, are 
parti-totally co-exclusive of each other. It was called by 
De Morgan the Gorgon Syllogism, alluding, I presume, to 
the petrifying effect it produces upon all mankind who are 
not lawyers. It is plainly in the mood E Y w, and though 
it does not appear in Thomson s table, may be considered 
a weakened form of E Y O, the seventh negative mood 
of the first figure. The point of the matter, however, is that 
Hamilton, in his later writings, proposed to depart from 
the Aristotelian sense of the mark of particular quantity 
some. As stated in his Lectures, Vol. IV. p. 281, the view 
which he wished to introduce is that some should mean 
some at most, some only, some not all. But, if we 
apply this meaning of some to the conclusion of the 



158 QUANTIFIED PREDICATE. [CHAP. xvm. 

Gorgon Syllogism, it produces the ridiculous result that, 
though lawyers are not stone, all the rest of mankind are stone. 
De Morgan is unquestionably correct, and this Gorgon 
Syllogism brings to ruin Hamilton s long adequately tested 
and matured system. 

The particulars of the discussion between De Morgan 
and Professor Spencer Baynes about this Gorgon Syllogism, 
and kindred matters, may be found in the Athenceum of 
1 86 1 and 1862 and elsewhere. 

It is curious that De Morgan states the Gorgon Syllo 
gism differently in the Athenaeum of 2d November, 1861, 
p. 582, and in his Fifth Memoir on the Syllogism, p. 10 ; 
but the difference is not material to the final issue. 

II. The month of May has no "R" in its name ; 
nor has June, July, or August : all the hottest 
months are May, June, July, and August : 
therefore, all the hottest months are without 
an " R " in their names. 

This is Whately s example No. 117, and as he refers the 
student to Book IV., Chap. I., i, which treats of induction, 
he evidently regards it as an Inductive Syllogism. It would 
have been referred by Hamilton to the Thomsonian mood 
E U E, the minor premise being treated as a doubly uni 
versal proposition. There can be no doubt, however, that 
the minor is really disjunctive, thus : A hottest month is 
either May, or June, or July, or August. The major is a 
compound sentence, comprising four separate propositions, 
May has no R in its name/ June has no R, etc. (See 
Elementary Lessons, Lesson XXV., p. 215.) 



CHAPTER XIX. 

EXERCISES ON THE QUANTIFICATION OF THE PREDICATE. 

i. EXPRESS carefully, in full logical form, with quantified 
subjects and predicates, the following propositions ; assign 
the Thomsonian symbol in each case : 

(1) Thoughts tending to ambition, they do plot unlikely 

wonders. 

(2) Fools are more hard to conquer than persuade. 

(3) Heaven has to all allotted, soon or late, 

Some happy revolution of their fate. 

(4) Justice is expediency. 

(5) This is certainly the man I saw yesterday. 

(6) Man is the only animal with ears that cannot move 

them. 

(7) Wisdom is the habitual employment of a patient and 

comprehensive understanding in combining various 
and remote means to promote the happiness of 
mankind. 

(8) It is among plants that we must place all the Dia- 

tomaceae. 

(9) When the age is in the wit is out. 

"(10) Every man at forty is either a fool or a physician, 
(n) Some men at forty are neither fools nor physicians. 
(12) Some men at forty are both fools and physicians. 



160 QUANTIFIED PREDICATE. [CHAP. 

(13) L fitat c est moi, as Louis the Fourteenth used to say. 

(14) There are no coins excepting those made of metal, 

if we overlook a few composed of porcelain, glass, 
or leather. 

(15) Antisthenes said Stlv Kraa-Bat vow rj fipo^ov. 

(16) All animals which have a language have a voice, but 

not all which have a voice have a language. 

(17) The elephant alone among mammals has a pro 

boscis. 

(18) Prudence is that virtue by which we discern what is 

proper to be done under the various circumstances 
of time and place. 

(19) Whatever is, is right. 

(20) There are arguments and arguments. 

(21) A dispute is an oral controversy, and a controversy 

is a written dispute. 

(22) There beth workys of actyf lyf othere gostiy othere 

bodily. 

(23) The only Roman who gave us a summary of 

Aristotle was the only Roman who gave us a 
summary of Euclid. 

(24) Zenobia declared that the last moment of her reign 

and of her life should be the same. 

(25) As it asketh some knowledge to demand a question 

not impertinent, so it requireth some sense to 
make a wish not absurd. 

(26) Mankind consists of dark men and fair men. 

(27) To say that Mr. Raffles was excited was only 

another way of saying that it was evening. 

(28) Though all well educated men are not discoverers, 

all discoverers are well educated men. 

(29) No man is esteemed for gay garments but by fools 

and women. 



EXERCISES. 161 

(3) Quand celui qui ecoute n entend rien, et quand celui 
qui parle n entend plus, c est la metaphysique. 

(31) Friendship finds men equal or makes them so. 

(32) I can fly or I can run. 

(33) A man is an ill husband of his honour that entereth 

into an action, the failing wherein may disgrace 
him more than the carrying of it through can 
honour him. 

(34) Scribendi recte sapere est principium et fons. 

(35) Tools are only simple machines, and machines are 

only complicated tools. 

(36) The wise man knows the fool, but the fool knows 

not the wise man. 

(37) It is scandalous that he who sweetens his drink by 

the gift of the bees, should by vice embitter 
Reason, the gift of the Gods. 

(38) A and B and C and D, etc., etc., wear black coats 

on Sundays ; in fact every man I know does so. 

(39) All the Apostles were Jews, because this is true of 

Peter, James, John, and every other Apostle. 

(40) A dose of arsenic is given to a living healthy dog. 

Soon after the dog dies. Arsenic is therefore a 
poison. 

2. How can a chain of reasoning, founded on circum 
stantial evidence, be represented in syllogistic form ? [E.] 

3. Having special regard to the logical sense of some, 
what do you think of the validity of the following argument 
(Thomson s Syllogistic Mood A E rj) ? 

All Fis some X; 
No Zis any K; 
Therefore, No Z is some X. 
M 



162 QUANTIFIED PREDICATE. [CHAP. 

4. We have been assured that "all X is some Y" is 
contradicted by "all Y is some X" a proposition which 
cannot be made good except by some being declared not 
all (De Morgan, Third Memoir, 1858, p. 24.) Investi 
gate this point. 

5. Take stone and solid as subject and predicate, 
and convince yourself that the proposition in o>, some 
stone is not some solid, cannot be contradicted by any 
propositions of the forms U, A, I, Y, E, O, >?, having the 
same subject and predicate. 

6. Write out the various judgments, including U and Y, 
which are logically opposed to the judgment, No puns are 
admissible. State in the case of each judgment thus formed 
what is the kind of opposition in which it stands to the 
original judgment, and also the kind of opposition between 
each pair of the new judgments. [c.] 

7. The judgment, "No birds are some animals," is never 
actually made because it has the semblance only, and not 
the power of a denial. Examine this statement. [p.] 

8. Draw inferences from the following : 

If Sir Thomas was imbecile, then Oliver was right ; and 
unless Sir Thomas was imbecile, Oliver was not 
wrong. [p.] 

9. Examine the following arguments in those which are 
false point out the nature and name of the fallacy ; arrange 
those which are valid syllogisms in the usual form, and give 
the symbolic description of the mood. 

(i) All the householders in the kingdom, except women, 
are legally electors, and all the male householders 
are precisely those men who pay poor-rates ; it 
follows that all men who pay poor-rates are 
electors. 



xix.] EXERCISES. 163 

(2) All the times when the moon comes between the 

earth and the sun, are the sole cases of a solar 
eclipse ; the 1 1 th of February is not such a time ; 
therefore, the nth of February will exhibit no 
eclipse of the sun. [THOMSON.] 

(3) All men are mortals, and all mortals are all those 

who are sure to die ; therefore, all men are all 
those who are sure to die. 

(4) The Claimant is unquestionably Arthur Orton : for 

he is Castro who is the same person as Arthur 
Orton. 

10. Which of the following moods are legitimate, and 
in what figures : E Y O, Y A A, Y A Y, I Y I, Y Y Y, 
A E E ? [M.] 

1 1. Examine the validity of the following moods : 

Figure I. Figure II. Figure III. 

U AU AAA YEE 
YOO AYY OYO [c.] 

12. Exemplify any of the following moods, and deter 
mine in how many figures each is valid : U U U, I U I, 
Y U Y, i\ U % 01 U co. 



CHAPTER XX. 

EXAMPLES OF ARGUMENTS AND FALLACIES. 

THIS chapter contains a large collection of examples of 
Arguments and Fallacies collected from many sources. They 
form additional illustrations and exercises to supplement 
what are given in the previous chapters. The student is to 
determine in the case of each example whether it contains 
a valid or fallacious argument. In the former case he is to 
throw the example into a regular form, and assign the 
technical description of that form, whether a mood of the 
categorical syllogism, or of the hypothetical or disjunctive 
syllogism, etc. In some examples two or more syllogisms, 
or two or more different forms of reasoning, will be com 
plicated together. They must of course be analysed and 
exhibited separately. 

When the existence of fallacy is suspected, the student 
must endeavour to reduce this to a distinct paralogism or 
breach of the syllogistic rules, exhibiting the pseudo-mood 
or pseudo-form of reasoning. In many cases, however, the 
fallacy may be of the kinds described in the Aristotelian 
text -books as Semi -logical or Material. These fallacies 
have been explained in the Elementary Lessons (Lessons 
XX. and XXL), but for convenience of reference a simple 
list of the kinds of Fallacies is given below. It has not 
been found practicable to undertake in this book a full 



CHAP, xx.] ARGUMENTS AND FALLACIES. 165 

exemplification of the subject of Fallacies. The student is 
therefore referred to the Elementary Lessons named, or to 
any of the following writings on the subject : 

De Morgan s Formal Logic, Chapter XIII., as amusing 
as it is accurate and instructive ; Whately s Logic, Book III., 
perhaps the best and most interesting part of this celebrated 
text-book ; Edward Poste s edition of Aristotle on Fallacies. 

Paralogisms. 

1. Four Terms. Breach of Rule I. 

2. Undistributed Middle. Breach of Rule III. 

3. Illicit Process of Major or Minor Term. Breach of 

Rule IV. 

4. Negative premises. Breach of Rule V. 

5. Negative Conclusion, from affirmative premises, and 

vice versa. Breach of Rule VI. 

Breaches of Rules VII. and VIII. can be resolved into 
one or other of the above. 

Semi-logical Fallacies. Material Fallacies. 

1. Equivocation. i. Accident. 

2. Amphibology. 2. Converse Fallacy of Accident. 

3. Composition. 3. Irrelevant Conclusion. 

4. Division. 4. Petitio Principii. 

5. Accent. 5. Non Sequitur. 

6. Figure of Speech. 6. False Cause. 

7. Many Questions. 

1. France, having a warm climate, is a wine-producing 
country. [E.] 

2. Livy describes prodigies in his history ; therefore he 
is never to be believed. [E.] 



166 ARGUMENTS AND FALLACIES. [CHAP. 

3. All the metals conduct heat and electricity ; for 
iron, lead, and copper do so, and they are (all) metals. 

[E.] 

4. A charitable man has no merit in relieving distress, 
because he merely does what is pleasing to himself. [E.] 

5. What is the result of all this teaching? Every day 
you hear of a fraud or forgery, by some one who might 
have led an innocent life, if he had never learned to read 
or write. [E.] 

6. The use of ardent spirits should be prohibited by law, 
seeing that it causes misery and crime, which it is one of 
the chief ends of law to prevent. [E.] 

7. Pious men only are fit to be ministers of religion; 
some ignorant men are pious ; therefore ministers of 
religion may be ignorant men. [L.] 

8. No punishment should be allowed for the sake of the 
good that may come of it ; for all punishment is an evil, 
and we are not justified in doing evil that good may come 
of it. [E.] 

9. We know that God exists because the Bible tells us 
so ; and we know that whatever the Bible affirms must be 
true because it is of Divine origin. [E.] 

10. The end of punishment is either the protection of 
society or the reformation of the individual. Capital 
punishment ought therefore to be abolished. It does not 
in fact prevent crimes of violence, and so fails to protect 
society, while on the other alternative it is absurd. [E.] 

11. The glass is falling; therefore we may look for 
rain. [E.] 

12. This is a dangerous doctrine, for we find it up 
held by men who avow their disbelief in Revelation. 

13. If there is a demand for education, compulsion is 
unnecessary. [E.] 



xx.] EXAMPLES. 167 

1 4. Actions that benefit mankind are virtuous ; therefore 
it is a virtuous action to till the ground. 

15. Slavery is a natural institution ; therefore it is wrong 
to abolish it. 

1 6. No fool is fit for high place; John is no fool; 
therefore John is fit for high place. [E.] 

17. He is not a Mahometan, for no Mahometan holds 
these opinions. [E.] 

1 8. Mind is active ; matter is not mind ; therefore 
matter is not active. [E.] 

19. He must be a Mahometan, for all Mahometans 
hold these opinions. [E.] 

20. If we are to believe philosophers, knowledge is 
impossible, for one set of them tell us that we can know 
nothing of matter, and another that we can know nothing 
of mind. [o.] 

21. Old age is wiser than youth; therefore we must be 
guided by the decisions of our ancestors. [o.] 

22. Political assassins ought not to be punished, for 
they act according to their consciences. [o.] 

23. If education is popular, compulsion is unnecessary; 
if unpopular, compulsion will not be tolerated. [o.] 

24. Nations are justified in revolting when badly 
governed, for every people has a right to good govern 
ment. [E.] 

25. These two figures are equal to the same figure, and 
therefore to each other. 

26. Opium produces sleep, for it possesses a soporific 
virtue. [E.] 

27. Wealth is in proportion to value, value to efforts, 
efforts to obstacles : ergo, wealth is in proportion to 
obstacles. 



1 68 ARGUMENTS AND FALLACIES. [CHAP. 

28. When Croesus has the Halys crossed, a mighty army 
will be lost. 

29. A manor cannot begin at this day, because a court- 
baron cannot now be founded. 

30. Poeta nascitur, non fit ; how absurd it is then to 
teach the making of Latin verses ! 

31. Aio te ^Eacida, Romanes vincere posse. 

32. Every rule has exceptions ; this is a rule, and there 
fore has exceptions ; therefore there are some rules that 
have no exceptions. [E.] 

33. All that perceives is mind; the existence of objects 
consists in being perceived ; therefore the existence of 
objects necessarily depends on mind. [E.] 

34. Some objects of great beauty merely please the eye ; 
for instance, many flowers of great beauty, and accordingly 
they answer no purpose but to gratify the sight. [H.] 

35. A miracle is a violation of the laws of nature; and, 
as a firm and unalterable experience has established those 
laws, the proof against a miracle, from the very nature of 
the fact, is as entire as any argument from experience can 
possibly be. [E.] 

36. The imagination and affections have a close union 
together. The vivacity of the former gives force to the 
latter. Hence the prospect of any pleasure with which we 
are acquainted affects us more than any other pleasure 
which we may own superior, but of whose nature we are 
wholly ignorant. [E.] 

37. Common salt consists of a metal and a metalloid, 
for it consists of sodium and chlorine, of which one is a 
metal, and the other a metalloid. [E.] 

3 8. The truth is, that luxury produces much good. A 
man gives half-a-guinea for a dish of green peas ; how much 
gardening does this occasion? (Dr. Johnson.) [o.] 



xx.] EXAMPLES. 169 

39. Nothing can be produced ; for what exists cannot be 
produced, as it is already in existence, and what does not 
exist cannot be produced, as, since it is not in existence, 
nothing can happen to it. [E.] 

40. The earth s position must be fixed, if the fixed stars 
are seen at all times in the same situations : now the fixed 
stars are not seen at all times in the same situations ; 
therefore the earth s position is not fixed. [E.] 

41. Protective laws should be abolished, for they are 
injurious if they produce scarcity, and they are useless if 
they do not. [E.] 

42. All who think this man innocent think he should 
not be punished; you think he should not be punished, 
therefore you think him innocent. [E.] 

43. If we are disposed to credit all that is told us, we 
must believe in the existence, not only of one, but of two 
or three Napoleon Buonapartes ; if we admit nothing but 
what is well authenticated, we shall be compelled to doubt 
the existence of any. How, then, can we be called 
upon to believe in the one Napoleon Buonaparte of 
history ? [o.] 

44. We cannot know what is false, for knowledge cannot 
be deceptive, and what is false is deceptive. [E.] 

45. A necessary being cannot be the effect of any cause ; 
for, if it were, its existence would depend upon the existence 
of its cause, and therefore would not be necessary. [E.] 

46. The table we see seems to diminish as we move 
from it ; but the real table suffers no change : it was not, 
therefore, the table itself, but only its image, that was 
present to the mind. 

47. The existence of sensations consists in being per 
ceived : all objects are really collections of sensations ; 
therefore their existence consists in being perceived. [E.] 



1 70 ARGUMENTS AND FALLACIES. [CHAP. 

48. If the earth were of equal density throughout, it 
would be about 2-^ times as dense as water : but it is about 
5^ times as dense ; therefore the earth must be of unequal 
density. 

49. Whatever is conditioned must depend on some cause 
external to itself: this world is conditioned by time and 
space; therefore this world depends upon some cause 
external to itself. [F,] 

50. It sometimes happens that an electrical current is 
excited, where none but magnetic forces are directly called 
into play ; for such a current, in certain cases, is excited in 
an electric non-conductor by moving a magnet to or away 
from it. [R.] 

51. The Quaker asserts that if men were true Christians, 
and acted upon their religious principles, there would be no 
need of armies. Hence he draws the conclusion that a 
military force is useless, and being useless, pernicious. 

52. Detention implies at least possession; for detention 
is natural possession. 

53. Nothing can be conceived without extension : what 
is extended must have parts ; and what has parts may be 
destroyed. [o.] 

54. Had an armistice been beneficial to France and 
Germany, it would have been agreed upon by those powers : 
but such has not been the case ; it is plain therefore that an 
armistice would not have been advantageous to either of 
the belligerents. [o.] 

55. By the law of nature as soon as Adam was created 
he was governor of mankind, for by right of nature it was 
due to Adam to be governor of his posterity. [o.] 

5 6. When men are pure, laws are useless ; when men are 
corrupt, laws are broken. 

57. There are many arguments which we recognise as 



xx.] EXAMPLES. 171 

valid which it is impossible to express in a syllogistic form : 
therefore the syllogism is valueless as a test of truth, [o.] 

58. No man can be a law to himself; for law implies a 
superior who gives the law and an inferior who obeys it ; 
but the same person cannot be both ruler and subject, [o.] 

59. It is injustice to the intellect of women to refuse 
them the suffrage ; for the reigns of many queens, as our 
own Elizabeth or Anne, have been famous for literary 
productions. [o.] 

60. To allow every man unbounded freedom of speech 
must be advantageous to the State, for it is highly conducive 
to the interests of the community that each individual 
should enjoy an unlimited liberty of expressing^ his senti 
ments, [o.] 

6 1. Your sorrow is fruitless, and will not change the 
course of destiny. Very true, and for that very reason I 
am sorry. [o.] 

62. Because some individuals have in their very child 
hood advanced beyond the youthful giddiness and debility 
of reason, it only needs a proper system of education to 
make other young people wise beyond their years. [o.] 

63. Haste makes Waste, and Waste makes Want ; there 
fore a man never loses by delay. [o.] 

64. If peace at any price is desirable, war is an evil ; and 
as war is confessedly an evil, peace at any price is de 
sirable, [o.] 

65. The two propositions, Aristotle is living, and, 
Aristotle is dead, are both intelligible propositions ; they 
are both of them true or both of them false, because all 
intelligible propositions must be either true or false. [E.] 

66. No form of democracy is subject to violent revolu 
tions, because it never excludes the mass of the people 
from political power. [E.] 



172 ARGUMENTS AND FALLACIES. [CHAP. 

67. The student of History is compelled to admit the 
Law of Progress, for he finds that Society has never stood 
still. [ E .] 

68. It is fated that I shall or that I shall not recover, 
in either of which cases the employment of a physician 
is useless, and therefore inexpedient. [E.] 

69. The assertion that men much occupied in public 
affairs cannot have time for literary occupations is disproved 
by such instances as Julius Caesar, Alfred, Lord Bacon, Sir 
G. C. Lewis, the Earl of Derby, Mr. Gladstone, and the 
late Emperor of the French. 

70. Whatever had a beginning in time has limits in 
space : the universe has no beginning in time : therefore 
the universe has no limits in space. 

7 1 . The mollusc is an aggregate of the second order ; 
for there is no sign of a multiplicity of like parts in its 
embryo. 

72. The farmers will not pay in rent more than the net 
produce of their farms ; for no trading class will continue a 
losing business. [L.] 

73. The knowledge of things is more improving than the 
knowledge of words. The study of Physics must therefore 
be more improving than the study of Languages. [E.] 

74. The moral world is far from being so well governed 
as the material ; for the former, although it has its laws, 
which are invariable, does not observe these laws so con 
stantly as the latter. [p.] 

75. England has a gold coinage and is a very wealthy 
country ; therefore, it may be inferred that other countries 
having a gold coinage must be wealthy. 

76. Most parents are the best judges of the age at which 
their children should be sent to school, and as it is un- 



xx.] EXAMPLES. 173 

desirable to interfere with those who are the best judges of 
their children s interest, it follows that parents should not 
be compelled to send their children to school. 

77. Among the bodies which do not move in elliptic 
orbits are some of the comets ; but all bodies which do 
move in elliptic orbits return periodically ; hence, some 
bodies which return periodically cannot be comets. 

7 8. Some rate-payers are clearly not fit for their duties ; 
for all male rate-payers are electors, and some electors who 
accept bribes are clearly unfit for their duty of electing 
representatives. 

79. Whatever is done skilfully appears to be done with 
ease ; and art, when it is once matured to habit, vanishes 
from observation. We are therefore more powerfully ex 
cited to emulation by those who have attained the highest 
degree of excellence, and whom therefore we can with the 
least reason hope to excel. [L.] 

80. It is absurd to maintain that when we cannot avoid 
thinking or conceiving a thing, it must be true ; for some 
persons cannot be in darkness without thinking of ghosts, 
in which they do not believe. [R.] 

8 1. How can any one maintain that pain is always an 
evil, who admits that remorse involves pain, and yet may 
sometimes be a real good ? [c.] 

82. The time is past in which the transmission of news 
can be measured by the speed of animals or even of 
steam ; for the telegraph is not approached by either. 

[DE MORGAN.] 

83. We enjoy a greater degree of political liberty than 
any civilised people on earth, and can therefore have no 
excuse for a seditious disposition. 

84. Those only who understand other languages are 
competent to treat correctly of the principles of their own ; 



174 ARGUMENTS AND FALLACIES. [CHAP. 

since such a competency requires a philosophical view of 
the nature of language in general. [L.] 

85. If matter must be merely phenomenal, I must be so 
too. [E.] 

86. A miracle is incredible, because it contradicts the 
laws of nature. [E.] 

87. There are no practical principles wherein all men 
agree, and therefore none which are innate. [E.] 

88. Potash contains a metal; for all alkalies contain a 
metal, and potash is an alkali. [E.] 

89. Quench not hope; for when hope dies, all dies. 

90. That is too bad : you have the impudence to say you 
are a materialist, while I know that you are a dancing 
master. 

9 1 . Blood is a colour ; for it is red, and red is a colour. 

92. Every incident in this story is very natural and prob 
able ; therefore the story itself is natural and probable. 

93. Dolor, si longus, levis ; si gravis, brevis : ergo, 
omnino fortiter sustinendus. [L.] 

94. Whether we live, we live unto the Lord; and 
whether we die, we die unto the Lord : whether we live 
therefore, or die, we are the Lord s. [Rom. xiv. 8.] 

95. Quand on n a point d amis, on n est pas heureux; les 
hommes faux et trompeurs n ont point d amis; ainsi les 
hommes faux et trompeurs ne sont pas heureux. 

96. Philippi was the city where the first Christian Church 
in Europe was founded : it was also the place where the 
republican army of Rome under Brutus and Cassius was 
finally defeated; hence the republican army was finally 
defeated at the city where the first Christian Church in 
Europe was founded. 

97. Switzerland is a republic, and, you will grant, a more 
stable power is not to be found ; nor, again, is any political 



EXAMPLES. 175 

society more settled than that of the United States. Surely, 
then, republican France can be in no danger of revolution. 

[R.] 

98. Expand into a syllogism, as briefly as you can, the 
argument contained in the dialogue of Shakespeare s King 
Henry VI. Part III. Act I. Scene i, between the words, 
my title s good, etc., and, succeed and reign. [o.] 

99. Quoniam deos beatissimos esse constat ; beatus 
autem sine virtute nemo potest : nee virtus sine ratione 
constare ; nee ratio usquam inesse nisi in hominis figura ; 
hominis esse specie deos confitendum est. [H.] 

100. Without order there is no living in public society, 
because the want thereof is the mother of confusion, where 
upon division of necessity followeth ; and out of division, 
destruction. [HOOKER, Ecclesiastical Polity, v. 8, s. i.] 

i o i. Whatever is contradictory to universal and invariable 
experience is antecedently incredible ; and as that sequence 
of facts which is called the order of nature is established, 
and in accordance with universal experience, miracles or 
alleged violations of that order are antecedently improbable. 

[E.] 

102. Justice is the profit of others; therefore it is un 
profitable to the just man to be just. r o 1 

103. In trade both buyer and seller profit : in the home 
trade both these profits remain in the country; in the 
foreign trade one profit goes to the foreign trader ; therefore 
the same population will be more profitably employed in 
the home than in the foreign trade. [o "I 

104. Whatever brings in money enriches. Hence the 
value of any branch of trade, or of the trade of the country 
altogether, consists in the balance of money it brings in ; 
and any trade which carries more money out of the country 
than it draws into it is a losing trade : and therefore, money 



176 ARGUMENTS AND FALLACIES. [CHAP. 

should be attracted into the country, and kept there by 
bounties and prohibitions. 

105. Distinction may be reasonably expected, because 
what is not uncommon may be reasonably expected, and 
distinction is not uncommon. [E.] 

1 06. Neither am I moved with envy; for if you are 
equal to, or less than myself, I have no cause for it ; and if 
you be greater, I ought to endeavour to equal you, and not 
to speak evil of you. [L.] 

107. Great men have been derided, and I am derided : 
which proves that my system ought to be adopted. 

[DE MORGAN, Paradoxes, p. 387.] 

1 08. Preventive measures are always invidious, for when 
most successful the necessity for them is the least apparent. 

109. Treason never prospers : What s the reason ? Why, 
when it prospers, none dare call it Treason. 

no. Neque quies gentium sine armis, neque arma sine 
stipendiis ; neque stipendia sine tributis habere queunt. 
[TACITUS, Hist. Lib. iv. cap. 74.] 

in. Men are not brutes; brutes are irrational: all 
irrational beings are irresponsible ; therefore, men are not 
free from responsibility. [H.] 

112. The best of all taxes are taxes on consumption and 
taxes on the transfer of property : now all the latter and 
many of the former are levied by stamps; stamp duties 
therefore are good taxes, and taxes on justice are all stamp 
duties ; therefore taxes on justice are good taxes. (See 
BENTHAM S Protest against Law Taxes, second edition, 

1816, pp. 53, 54-) 

113. Dr. Johnson remarked that a man who sold a 
penknife was not necessarily an ironmonger. What is the 
name and nature of the logical fallacy against which this 
remark was directed? 



xx.] EXAMPLES. 177 

114. When Columbus made the egg stand on end by 
breaking it, what fallacy may he be said to have committed? 

115. Either all things are ordered by an intelligent Being 
who makes the world but one family (and if so, why should 
a part, or single member complain of that which is designed 
for the benefit of the whole?); or else we are under the mis 
rule of atomes, and confusion. Now, take the case which 
way you please, there s either no reason or no remedy for 
complaint ; and therefore it is to no purpose to be uneasie. 

[Marcus Antoninus Meditations, ix. 40.] 

1 1 6. Silk is dearer than wool, and wool than cotton ; 
therefore silk is dearer than cotton. 

117. One napoleon is worth twenty francs, and twenty 
francs are worth about sixteen shillings ; therefore one 
napoleon is worth about sixteen shillings. 

1 1 8. The Prince of Wales is the eldest son of the reign 
ing sovereign ; and the eldest son of the reigning sovereign, 
if there be such, is the heir to the throne ; therefore the 
Prince of Wales is heir to the throne. 

119. It is a mistake to suppose that the rents paid to 
landlords are a burden on the public, since corn would not 
be more plentiful or cheaper if they were abolished. 

120. The Greeks have little respect for the petty honesty 
of small tradesmen ; we do not greatly admire the wiles of 
Ulysses ; therefore any common inward standard of morals 
is impossible. 

121. Dissent always weakens religion in the people ; for 
it sets itself in opposition to the National Church. 

122. We are not inclined to ascribe much practical value 
to that analysis of the inductive method which Bacon has 
given in the second book of the Novum Organum. It is 
indeed an elaborate and correct analysis. But it is an 
analysis of that which we are all doing from morning to 

N 



178 ARGUMENTS AND FALLACIES. [CHAP. xx. 

night, and which we continue to do even in our dreams. 
(MACAULAY, Essay on Bacon], [E.] 

123. The Claimant has undoubtedly many peculiarities 
of gait and manner which were characteristic of the missing 
baronet. Are not these therefore proofs of identity equi 
valent to the evidence of imposture afforded by the 
absence of tattoo-marks which the genuine man is proved 
to have possessed ? 

124. Even if it could be shown that animals perform 
certain actions which men could only perform by the aid of 
reason, it would by no means necessarily follow that animals 
perform them by its aid. [c.] 

125. If there s neither Mind nor Matter, 
Mill s existence, too, we shatter : 

If you still believe in Mill, 
Believe as well in Mind and Matter. [E.] 
126. If we accept Aristotle s testimony, we may infer 
that Anaximander was not one of the Ionian philosophers 
that accepted as the One material principle a mean term 
between Water and Air ; for, in the Physics, we read that 
he held the substances in nature to have been produced 
from the primordial principle by a process of secretion and 
not by a process of condensation and rarefaction ; while in 
the De Ccelo it is stated that other mode of production 
than the last-named was not put forward by any who 
adopted such a mean term for their principle. What 
syllogistic form (figure and mood) does this inference most 
naturally assume ? [ R ] 



CHAPTER XXI. 

ELEMENTS OF EQUATIONAL LOGIC. 

i. THE symbols employed in this system are the 
following : 

A, B, C, or other capital letters, signify qualities, or 
groups of qualities, forming the common part, or intensive 
meaning, of terms, or names of objects and classes of 
objects. 

a, b, c, or other small italic letters, are the corresponding 
negative terms ; thus a signifies the absence of one or more 
of the qualities signified by A. This notation for negatives 
was proposed by De Morgan (Formal Logic, p. 38). The 
mark = is the sign of Identity of Meaning of the terms 
between which it stands ; thus A = B indicates that the 
qualities signified by A are identical with the qualities 
signified by B. 

The sign ) signifies unexdusive alternation, including 
the ordinary meanings of both the conjunctions or and and. 
Thus A [ B means the qualities of A or those of B, or 
those of both A and B, if they happen to coincide. 

Juxtaposition of two letters forms a term whose meaning 
is the sum of the qualities signified by the two letters : 
thus AB means a union of the qualities of A and B. 



,8o EQUATIONAL LOGIC. [CHAP. 

2. The Laws of Combination of these symbols are as 
follow : 

The Law of Commutation. AB - BA : that is to say, 
the sum of qualities of A and B is evidently the same as 
the sum of qualities of B and A. The way of arriving at 
the sum may be different, but the result is identical. 

The Law of Simplicity. AA - A : if we have the 
same qualities twice over we get the same as if we named 
them once. 

The Law of Unity. A -|- A = A : the qualities of A 



J the qualities of A are simply the qualities of A. 
or 
The Law of Distribution. A(B -|- C) - AB -|- AC. 

The qualities of A with those of B ~ those of C are the 

same as those of AB - - those of AC. 
or 

The Law of Indifferent Order. B -|- C = C -|- B, which 
is sufficiently evident. 

3. The Laws of Thought are the foundation of all 
reasoning, and may thus be symbolically stated : 

The Law of Identity .... A - A. 

The Law of Duality or of ) AB ^ 

Excluded Middle . . . } 
The Law of Contradiction . Aa - o. 

The successive application of the Law of Duality to two, 
three, four, five or more terms, gives rise to the development 
of all possible logical combinations, called the Logical 
Alphabet, the first few columns of which are given below. 
The combinations for six terms are given in the Principles 
of Science, p. 94 (first ed. Vol. I. p. 109). 



xxi.] ELEMENTS. 181 

THE LOGICAL ALPHABET. 

i. ii. in. iv. v. vi. vi. continued. 



AX AB 


ABC 


A B C D 


ABCDE 


BCDE 


a X Kb 


KB c 


AB C d 


ABCD* 


B CD e 


a B 


Kb C 


A B c D 


ABCaTE 


B C^E 


a b 


Kb c 


A B c d 


AB Cde 


aE C d e 




/r T-? C* 


Kb C D 


ABcDE 


a E cD E 




aE c 


A3 C rf 


A B cDe 


a E c D 




ab C 


Kb c D 


AB cdE 


aE c d E 




a b c 


Kb c d 


A B c d e 


B c d e 






aE C D 


A-5CDE 


a b C D E 






B C rf 


KbC De 


a* C D e 






B c D 


KbC dE 


ab Cd E 






B * rf 


K b C d e 


ab C d e 






a b C D 


K b c D E 


ab c D E 






a b C rf 


Kb c D e 


c D ,? 






a 3 c D 


Kb c d E 


*Z (y <7 ^/ E 






abed 


Kb c d e 


ab c d e 



4. The one sole and all sufficient rule of inference is the 
following RULE OF SUBSTITUTION. 

FOR ANY TERM SUBSTITUTE WHAT IS STATED IN ANY 
PREMISE TO BE IDENTICAL IN MEANING WITH THAT TERM. 

The term may consist of any single letter, any juxtaposed 
letters, or any group of alternatives connected by the 
sign ] , the sign of unexclusive alternation. 

5. It is assumed as a necessary law that every term must 
have its negative. This was called the Law of Infinity in 
my first logical essay (Pure Logic, p. 65 ; see also p. 45); but 
as pointed out by Mr. A. J. Ellis, it is assumed by De Morgan, 
in his Syllabus, article 1 6. Thence arises what I propose to 
call the CRITERION OF CONSISTENCY, stated as follows : 

Any two or more propositions are contradictory when, and 
only when, after all possible substations are made, they 
occasion the total disappearance of any term, positive or 
negative, from the Logical Alphabet. 

The principle of this criterion was explained in p. 65 of 
the Essay on Pure Logic referred to, but subsequent inquiry, 
and the writings of Mr. A. J. Ellis, have tended to show 
the supreme importance of the criterion. 



182 EQUATIONAL LOGIC. [CHAP. 

The processes of this equational system of Logic are 
fully treated in the first seven chapters of the Principles of 
Science, and they are now amply illustrated by the problems 
which follow. 

6. How do you express in the new logic the four 
Aristotelian forms of proposition indicated by 
the vowels A, E, I, and O ? 

The answer is 

A. Every A is B. A = AB (i). 

E. No A is B. A - Kb (2). 

I. Some A is B. CA = CAB (3). 

O. Some A is not B. CA = CA (4). 

The first expresses the coincidence of the class A with 
part of the class B, namely A B, which is the equational 
mode of asserting that the A s form part of the B s. The 
second expresses similarly that the A s are found among 
the not-B s. In the third form some is expressed by the 
symbol C, and the proposition asserts that some A s (C A) 
are identical with a part of the class B. Some difficulties 
may arise about this form, owing to the ambiguity of the 
Aristotelian some, as elsewhere discussed (pp. 151-158). 
The fourth proposition is evidently the negative form of 
the third. 

7. How shall we express equationally the asser 
tion of Hobbes (De Corporc Politico, I. i. 13), 
that Irresistible might in the state of nature 
is right ? 

* Might is the principal part of the subject, but it is 
qualified or restricted in this proposition by the adjective 



XXL] QUESTIONS AND ANSWERS. 183 

irresistible, and by the adverbial in the state of nature. 
Thus putting 

A = irresistible ; C = in the state of nature ; 
B = might : D = right. 

The 1 subject is clearly ABC; and D is affirmed of it. 
But Hobbes cannot, of course, have meant that all right 
is irresistible might ; only in the state of nature is this true. 
As, indeed, irresistible might must overcome everything 
opposed to it, there can be nothing else right in the sphere 
of its action, so that the proposition would seem to have 
the form A B C = C D. It is not easy to be sure of the 
meaning even of Hobbes. 

8. Represent the meaning of the sentence Man 
that is born of a woman, is of few days, and 
full of trouble. 

The relative clause, that is bom of a woman, is 
evidently explicative, and we cannot suppose that there 
are any men not born of a woman. Hence taking 

A = man ; C = of few days ; 

B = born of a woman ; D = full of trouble ; 

the meaning seems to be expressed in the two pro 
positions 

A = AB; 

A = ACD. 

But if we may treat the sentence as an imperfectly expressed 
syllogism namely, because man is born of a woman he 
is, etc. then the premises obviously become A = AB, 
and B = B C D, and the conclusion by substitution for B 
in the first of its value in the second, is A = ABCD. 



1 84 EQUATIONAL LOGIC. [CHAP. 

p. Show how to obtain equationally the contra- 
positive of A = AB. 

This is explained in the Principles of Science, p. 83 
(first edition, Vol. I. pp. 97-102), but may be thus briefly 
repeated 

By the Second Law of Thought 

b = Kb -|. ab. 
Substitute for A its equal AB. 

b = AB -I- ab = o -|- ab, 
or, b = ab. 

Concerning the contrapositive see above, pp. 32, 
43-47, etc. 

10. Show how to obtain the complete contra- 
positive of A = B. 

As before 

b = Kb -|- ab = B# ] ab = o -|- ab = ab \ 
similarly 

a aB .|. ab = aK -|- ab = o -| ab = ab. 
Having now the two propositions 

a ab = b, 

it is plain that we may eliminate ab, and get 
a = b. 

11. What descriptions of the terms glittering 
thing and * not gold can you draw from the 
following assertions ? 

(1) Brass is not gold ; 

(2) Brass glitters. 

Let A = brass ; B = golden ; 

C = glittering thing. 
The premises are 

(i) A = Kb; (2) A = AC. 



XXL] QUESTIONS AND ANSWERS. 185 

Obviously C = ABC -|. KbC -|- aEC -|- abC. 
The first of the alternatives ABC is negatived by (i) ; 
but the second and fourth coalesce, and we have 

C = bC -|. tfBC; 

that is, a glittering thing is either not golden, or else it is 
golden, and then not brass. 

For b we similarly get 

b = bC -I- abc. 

Show that we may also infer 

C = C (a -|. b\ 

and b = b (a -|- C). 

12. How shall we represent in the forms of 
Equational Logic the moods of the old syllo 
gism ? 

All of the moods without exception may be solved by 
the indirect method, that is by working out the combinations 
consistent with the premises. Most of the moods may, how 
ever, be solved also by direct substitution, as will be seen 
in the following examples : 

Barbara. 

All men are mortal; (i) B - BC. 

All kings are men; (2) A = AB. 

. . All kings are mortal ; (3) A = ABC. 

We get (3) by substituting for B in (2), its equivalent 
BC in (i). The conclusion amounts to saying that king 
is equivalent to * king man mortal. If desired, we can 
by further substitution of A for A B in (3) obtain A = AC, 
or king = king mortal, which is a precise expression for 
the Aristotelian conclusion All kings are men. 



1 86 EQUATIONAL LOGIC. [CHAP. 

Celarent. 

No men are perfect ; ( i ) B = B<r. 

All kings are men; (2) A = AB. 

No kings are perfect ; (3) A = ABr. 

Solved, as in the last case, by direct substitution in (2) 
of the value of B given in (i). 

Darii. 

All mathematicians have well-trained ~| _ ~, 

intellects ; j 

Some women are mathematicians ; AB = ABC. 

Some women have well - trained ") A-orr* 

c AJJ AijC^U. 
intellects. ) 

Here A stands for the indefinite adjective some, and B 
for women, and we then treat AB as an undivided term, 
and obtain the result by direct substitution, exactly as in 
the previous moods. 

Ferio. 

No foraminifera are fresh - water \ _ 

inhabitants ; j 

Some components of chalk are fora- | 

. . ., C AJj = A-DV_x. 

mimfera ; ) 

Some components of chalk are not ) . 

.... > AB = ABG/. 

fresh-water inhabitants. j 

Except that a negative term d takes the place of the 
positive term D, in the last mood, there is no difference in 
form between them. In fact, all the four moods of the 
first figure present so great similarity that they may be said 
to be of one form of inference. 



xxi.] QUESTIONS AND ANSWERS. 187 

Cesare. 

The absolute is not phenomenal ; (i) C = Cb. 

All known things are phenomenal ; (2) A = AB. 
All known things are not the absolute. (3) A = ABr. 

We cannot by any direct substitution obtain the conclusion 
from the premises, as B appears in (2) and b in (i). But 
we may take the contrapositive of (i) as described before 
(p. 184), namely, B = B<r, and substitution in the second 
premise is then practicable. 

Camestres. 

All laws of nature are invariable ; (i) C = CB. 

No human customs are invariable; (2) A = Ab. 
No human customs are laws of nature. (3) A = Ac. 

As in the last mood, we must take the contrapositive of 
(i), namely b = b c, and substitute thereby in (2). 

13. The equational treatment of the moods Camestres, 
Cesare, and Camenes is described also in the Principles of 
Science, pp. 84-6 (first edition, Vol. I. pp. 99-101), or in 
the Substitution of Similars, pp. 47-49; but the following 
is the briefest way of getting the Aristotelian conclusion, 
of Camestres, as suggested by Mr. W. H. Brewer, M.A. 

Let the premises be 

(1) A = AC. 

(2) B - "Re. 

Multiply together, and we get 

AB = ABO- = o. 
Thus there is no such thing as AB. 
Similarly with Camenes. 



i88 EQUATIONAL LOGIC. [CHAP. 

14. The remaining moods need only be symbolically 
represented. In every case D = some : 



Major Premise. 


Minor Premise. 


Conclusion. 


Festino . 


. A = 


Kb . 


CD 


= BCD. 


CD = 


tfBCD 


Baroko 


. A = 


AB . 


CD 


- CD . 


CD = 


abCV 


Darapti 


. B = 


AB . 


B 


= CB . 


AB = 


CB 


Disamis . 


. BD = 


ABD . 


B 


= BC . 


BD = 


ABCD 


Datisi . 


. B = 


AB . 


BD 


= BCD . 


BCD = 


ABD 


Felapton . 


. B = 


aE . 


B 


= BC . 


BC = 


aE 


Bokardo . 


. BD - 


tfBD . 


B 


= BC . 


BCD = 


tfBCD 


Ferison . 


= 


aE . 


BD 


= BCD . 


BCD = 


tfBD 


Bramantip 


. A = 


AB . 


B 


= BC . 


ABC = 


A 


Camenes . 


. A = 


AB . 


B 


= Br . 


C = 


aC 


Dimaris . 


.AD = 


ABD. 


B 


= BC . 


AD - 


ABCD 


Fesapo 


. A - 


Kb . 


B 


= BC . 


BC = 


<zBC 


Fresison 


A = 


Kb 


BD 


- BCD . 


BCD = 


0BCD 



15. Exhibit the logical force of the motto adopted 
by Sir W. Hamilton 

(1) In the world there is nothing great but 
Man. 

(2) In Man there is nothing great but Mind. 



Let A - in the world ; 
B = man ; 



C = possessing mind ; 
D = great. 



The conditions may be represented as 

(i) A = ABD -I- Kbd. 
(a) B = BCD -I- 



As it may be understood, though unexpressed, that 



XXL] QUESTIONS AND ANSWERS. 189 

(3) all men are in the world, and that (4) all possessing 
mind are men, we have further 

(3) B = AB. (4) C = CB. 

The combinations are thus reduced to 

ABCD abcD 

hJbcd abed. 

Observe that, if mind were regarded not as an attribute 
but as a physical part of man, we could not treat the 
assertion as one of simple logical relation. 

16. What is the meaning of the assertion that 
All the wheels which come to Croyland are 
shod with silver ? 

If we take 

A = wheel ; 

B = coming to Croyland ; 

C = shod with silver, 

the assertion, as it stands, is evidently in the form 
AB = ABC (i). 

But it was always understood, no doubt, that this adage 
was to be joined in the mind with the tacit premise No 
wheels are shod with silver, expressed by 

A = Kc (2). 

There seems, at first sight, to be contradiction between 
these premises ; for (i) speaks of wheels shod with silver 
and (2) denies that there are such things. The explanation 
is obvious, namely, that there are no such things as wheels 
coming to Croyland. Of the four combinations containing A, 



igo EQUATIONAL LOGIC. [CHAP. 



is negatived by (i), and ABC and AC by (2), so that 
the description of A is given thus, 
A = Kbc, 
or, by substitution of A for Ac, 

A = A, 

that is, no wheels come to Croyland. This is of course 
the inference which the adage was intended to suggest, 
Croyland being an ancient abbey lying among the fens of 
Cambridgeshire, where in former days no wheeled vehicle 
could make its way. 

This question illustrates the important logical principle 
that all propositions ought, strictly speaking, to be interpreted 
hypothetically. We have only to put these premises in the 
hypothetical form, and we see that they make a reasonable 
destructive hypothetical syllogism thus 

If any wheels come to Croyland they are shod with 
silver ; but no wheels are shod with silver ; therefore, 
no wheels do come to Croyland. 

17. Ruminant animals are those which have 
cloven feet, and they usually have^ horns ; the 
extinct animal which left this footprint had a 
cloven foot; therefore it was a ruminant animal 
and had horns. Again, as no beasts of prey 
are ruminant animals it cannot have been a 
beast of prey. 

The above problem is given in the Elementary Lessons 
(p. 321, No. 78). Taking our symbols thus 

A = ruminant ; D = extinct animal ; 

B = having cloven feet ; E = beast of prey ; 

C = having horns ; 
we have clearly A = B (i). 



XXL] QUESTIONS AND ANSWERS. 191 

The statement that ruminant animals usually^ have horns 
may be formalised as 

BA = BAG (2), 

that is to say, a certain particular portion of the class A, 
BA have horns. Next we have 

D = DB (3). 

Substituting in (3) by (i) we get 

D = DA (5); 

showing that the extinct animal was a ruminant. But, as 
we cannot substitute between (3) and (2), it is erroneous to 
assert that it had horns. If by usually we mean in the far 
greater number of cases, then there is a considerable prob 
ability, but no certainty that it had horns. 

Again, we have as an additional premise, that beasts of 
prey are not ruminant, or 

E = ^E (4), 

which, taken with 

D = DA (5), 

our previous conclusion, gives a syllogism in Cesare, estab 
lishing that D is not E, or that the extinct animal cannot 
have been a beast of prey. This we might sufficiently prove 
symbolically by multiplying the respective members of (4) 
and (5) together, giving 

DE = #EDA = o. 

This shows that inconsistency arises from supposing that 
this D can be also E. The same result might be worked 
out by combinations, giving D = D<?. 

1 The first edition of the Lessons reads always instead of usttally. 



i 9 2 EQUATIONAL LOGIC. [CHAP. 

18. Take the proposition All crystals are solids/ 
and ascertain precisely what it affirms, what it 
denies, and what it leaves doubtful. 

[A. J. ELLIS.] 
Taking A = crystal, and B = solid, the proposition is 

in the form 

A = AB. 

The conceivable combinations are four in number, 

namely, 

AB, A, #B, and ab. 

Of these, only Ad is inconsistent with the premise, that is 
to say, the premise All crystals are solids denies the 
existence of such things as * unsolid crystals. We cannot 
strike out AB, because then there would be no such thing 
as crystals left ; hence the premise affirms the existence of 
solid crystals, in the sense that any other proposition deny 
ing that crystals are solid, or that solids may be crystals, 
would stand in contradiction to our premise. 

Again, we may not strike out a b, because there would 
then be no such thing as b, or not-solids. Hence to avoid 
contradiction of our premise, there must be such a thing 
as c a non-crystal which is not-solid. If we are to hold to 
our adopted Criterion of Consistency (p. 181), we must 
say that at least one case of ab exists, so that to avoid self- 
contradiction, some, that is at least one case of not-crystal, 
must be allowed to exist l and to be not-solid. This con 
firms the conclusion which we previously obtained by the 
Aristotelian logic from the same premise (p. 48). But the 
combination 0B may be removed or left without affecting 
the truth of the premises, which therefore leave it entirely 
in doubt whether not -crystals which are solids exist. 
1 Concerning the logical sense of the verb exist, see pp. 141-2. 



XXL] QUESTIONS AND ANSWERS. 193 

Not -crystals, not -solids must exist, but Not crystals- 
solids, may or may not. But if they do not, then crystals 
and solids will be coincident classes. 

To sum up 

A = AB affirms that all As are Bs ; 

A = AB affirms that all not-Bs are not-As ; 

A = AB affirms that some not-As are not-Bs ; 

A = AB denies that all not-As are not-Bs ; 

A = AB leaves doubtful whether not-A can be B. 

Ip. Are the following propositions equivalent 
each to the other ? 

(1) All who were there talked sense ; 

(2) All who talked nonsense were away. 

[DE MORGAN.] 
Putting 

A = being there ; a = not being there ; 

B = talking ; b = not talking ; 

C = sensible ; c = not sensible ; 

the first premise seems to mean that all who were there 
(A), talked and were sensible ; that is 

A = ABC (i). 
The second is to the effect 

B* = a&c (2) ; 

that is to say, those who talked (B), but not sensibly 
(c\ were away (a). Now (i) negatives three combinations, 
AB<r, A<5C, and Kbc\ whereas (2) negatives only AB<r. They 
are, therefore, very different propositions; for (2) allows 
that some may have been present who did not talk at all 
(b\ whether sensible people or not (C or c). Nevertheless, 
there is this much relation between the two propositions, 
that we can infer (2) from (i). If all who were there talked 

o 



i 9 4 EQUATIONAL LOGIC. [CHAP. 

sense, those people who talked nonsense, assuming there to 
be such persons, must have been absent. But we cannot 
invert this relation. Because those who talked nonsense 
were away, it does not follow that those who were present 
talked sense ; they may all have been silent. 

20. De Morgan says (Syllabus, p. 14), Any one 
who wishes to test himself and his friends upon 
the question whether analysis of the forms of 
enunciation would be useful or not, may try 
himself and them on the following question : 
Is either of the following propositions true, and 
if either, which ? 

(1) All Englishmen who do not take snuff 

are to be found among Europeans who 
do not use tobacco. 

(2) All Englishmen who do not use tobacco 

are to be found among Europeans who 
do not take snuff. 

Required immediate answer and demonstra 
tion. 
Assigning symbols as follows : 

A = Englishmen ; C = taking snuff; 

B = Europeans j D = using tobacco ; 

it is pretty obvious that the above propositions are thus 
symbolised 

(1) Ac = 

(2) M = 

We are to compare these with the well-known relations 
of the terms, which may be assumed to be 

(3) A = AB ; 



xxi.] QUESTIONS AND ANSWERS. 195 

that is, * Every Englishman is an European, and 

(4) C = CD; 

that is, All who take snuff use tobacco. Now, in working 
out the combinations, we find that the class Ac is composed 
under conditions (3) and (4) as follows : 
A^ = ABd3 -I- ABrt 

The truth is, then, that Englishmen who do not take snuff 
consist of English Europeans not taking snuff, but using 
tobacco, and of English Europeans neither taking snuff nor 
using tobacco. In short (i) is erroneous in ignoring the 
fact that some Englishmen not using snuff may be Europeans 
who do use tobacco for smoking. 

According to assumption (2) the description of h.d is 
AIW, which coincides with the description drawn from 
(3) and (4). Thus it is true that all Englishmen who do 
not use tobacco are to be found among Europeans who do 
not take snuff ; the negation of the larger term, using tobacco, 
includes the negation of the narrower one, using snuff. 
But it by no means follows that because our inference about 
Ad is the same from (2) as from (3) and (4), therefore these 
conditions are identical, as will be seen in the following 
descriptions of the class A as furnished under the several 
suppositions and conditions 

(1) A - ABC ! AEcd -I- 

(2) A = ABCD i AB<r -I- 
(3) and (4) A = ABCD -|- AB*. 

21. What can we infer about the term Europeans 
from the following premises ? 

(1) All Continentals are Europeans ; 

(2) All English are Europeans ; 

(3) No English are Continentals. 



I9 6 EQUATIONAL LOGIC. [CHAP. 

Taking A, B, C to represent Continentals, English, and 
Europeans respectively, the premises become 

(1) A - AC. 

(2) B = BC. 

(3) B = 0B. 

The combinations left uncontradicted are the four 
AC, 0BC, abC, abc, whence we learn that Europeans, 
C, consist of Continentals who are not English, of English 
who are not Continentals, and of any others, who are neither 
Continentals nor English (abC). 

22. Criticise Thomson s Immediate inference 
by the sum of several predicates. . . . From 
a sufficient number of judgments in A, having 
the same subject, a judgment in U may be 
inferred, whose predicate is the sum of all the 
other predicates. 

This question has been answered in the Principles of 
Science, p. 61 (first ed. Vol. I. p. 73). Judgments in A are 
of the form P = PQ, P = PR, P = PS, etc, and by 
summing up the predicates by successive substitution in the 
second side of P - PQ, we may get P == PQRS 
But this does not give a proposition of the form U which, 
as described by Thomson, is of the form P = X. 

23. Represent the following argument from 
Thomson s Laws of Thought, 107 : 

All P is either C or D or E ; 
S is neither C nor D nor E ; 
therefore, S is not P. 



xxi.] QUESTIONS AND ANSWERS. 197 

The premises are respectively : 

P - PC -I- PD -I- PE. 
S = Scde. 

We get the conclusion in the briefest way by multiplying 
the two premises together as they stand ; thus : 

PS = P (C ! D -I- E) Scde = o -|- o .|. o. 

Each alternative is found to be contradictory, so that 
there is no such thing as PS, that is to say, no P is S. 

The argument is not, however, correctly described by 
Dr. Thomson as in the syllogistic mood U E E, nor are the 
other forms of argument given in the same section syllo 
gistic. They are disjunctive in character. 

24. If Abraham were justified, it must have been 
either by faith or by works : now he was not 
justified by faith (according to James), nor by 
works (according to Paul) : therefore Abraham 
was not justified. [w.] 

There is some difficulty in deciding on the best method 
of symbolising this argument, owing to the vagueness of 
the conditions when analysed ; but the following seems to 
be the best representation : 

Let A = Abraham ; C = justified by works ; 

B = justified ; D = justified by faith. 

Then the premises are : 

AB - AB (C -I- D). 
A = Ac. 
A = Ad. 



198 EQUATIONAL LOGIC. [CHAP. 

These premises will be found to erase all the combina 
tions of A excepting hbcd, which gives the conclusion. 
The combinations of a are altogether unaffected and need 
not be examined. The student may try other modes of 
representing the premises, but should get A = Kb by every 
method. 



25. It must be admitted, indeed, that (i) a man 
who has been accustomed to enjoy liberty 
cannot be happy in the condition of a slave : 
(3) many of the negroes, however, may be 
happy in the condition of slaves, because 
(2) they have never been accustomed to enjoy 
liberty. [w.] 

Let A = man accustomed to enjoy liberty ; 
B = happy in condition of slave ; 
C = certain negroes. 

The premises may be stated in the forms 

A - Kb. (i) 

C = 0C. (2) 

The supposed conclusion is C = CB. (3) 

The possible combinations are as in the margin, from 



Me 



abC 
abc 



which it will be seen that 

C = tfBC ! abC; 
that is, are either B, happy, or b, not 
happy. 



The fallacy is that of Negative Premises or of Illicit 
Process of the Major. 



XXL] QUESTIONS AND ANSWERS. 199 

26. If that which is devoid of heat and devoid 
of visible motion is devoid of energy, it 
follows that what is devoid of visible motion 
but possesses energy cannot be devoid of 
heat. 

Let A = possessing heat ; 

B = possessing visible motion ; 
C = possessing energy, 

the universe being things unexpressed, and devoid of 
being taken as the negative of possessing. Then the 
condition is : 

ab = abc. 

By contraposition we obtain, using Mr. MacColl s notation 
for the negative of ab (See Preface) : 

C = C (ab) 

= C (A3 -I- a~B -I- AB) 
Hence bC = 



two self -contradictory alternatives disappearing. It can 
also be readily shown that this inference is equivalent to 
the original condition. 

27. Prove the logical equivalence of the pro 
position B .= AC ] ac and b = Ac [ aC. 

This might be shown by receding to the combinations of 
the Logical Alphabet, but it is more neatly proved by 
equating the negatives of each member of the first equation. 
If M = N, then also m n (p. 184); hence the negative 
of B must be identical with the negative of AC ! ac. 
Now the negative of B is b ; that of the compound and 



200 EQUATIONAL LOGIC. [CHAP. 

complex member is the compound of the negatives of the 
two alternatives. In Mr. MacColl s notation 

(AC -I- ac) = (AC) (ac) 

- (flC "I" Ac ! ac) (Ac -I- *C I AC). 

On multiplying out, the nine products are found to be all 
self-contradictory excepting 0C I* Ac, which is therefore 
the expression for b. Vice versa the negative of Ac -\- aC 
will be found to be AC -|- ac, so that the propositions are 
clearly equivalent. 

28. If no A is BC, what can I infer about the 
relation of B and AC ? 

The condition is 

A = Ab -I- Ac. 

Substitute in either side of 

ABC = ABC, 
and we get 

ABC - ABZ>C -I- ABG: - o, 

or B cannot be AC. 

29. It is known of certain things that (i) where 
the quality A is, B is not ; (2) where B is, and 
only where B is, C and D are. Derive from 
these conditions a description of the class of 
things in which A is not present, but C is. 

The premises are clearly : 

(1) . . A = Ab; 

(2) . . B = CD 



xxi.] QUESTIONS AND ANSWERS. 201 

The conceivable constituents of the class which is C 
but not-A are 

aC = tfBCD -I- aECd -|- abCD >\- abCd. 

Substituting CD for B in the second, and B for CD in 
the third alternative, we find that these combinations give 
contradictory results, namely aBCDd and 0BCD. It 
follows that 

aC = 0BCD -I- abCd. 

Observe that the premise (i) has no connection with this 
result, which is deducible from (2) alone. 

30. It has been observed that in a certain class 
of substances, (i) where the properties A and 
B are present, the property C is present ; and 
(2) where B and C are present, A is present. 
Does it follow that B is present where C and 
A are present ? 

The premises are obviously : 

(1) . . AB = ABC, 

(2) . . BC - ABC, 

which are equivalent to the single proposition 

AB - BC. 
The answer is obtained at once from I ABC 



the combinations in the margin, which 



show that 



AC = ABC -I- KbC ; 



that is, in the presence of A and C, B 
is indifferently present or absent. [Dr. 
Macfarlane (Algebra of Logic, 1879, p. 141), who gives this 



KbC 
Ate 



abC 

abc 



202 EQUATIONAL LOGIC. [CHAP. 

problem, requires more than half a page to solve it, using, 
moreover, sundry impossible logical fractions.] 

31. Given (i) that everything is either B or C, 
and (2) that all C is B, unless it is not A : 
prove that all A is B. [c.] 

The first condition is expressed by the assertion that 
every not-B is C, which carries with it the equivalent that 
every not-C is B. Thus we have 

b = bC. (i) 

The second assertion is less easy to interpret, because we 
are not told what happens if it, that is C, is not-A. The 
meaning appears, however, to be that C if it is A must 
be B, that is 

AC = ABC. (2) 

These conditions give the combinations 
ABC. 



abC. 

from the first two of which we learn that A is always B. 

32. If we throw every A is B into the form 
every A is B or B/ we have every A which is 
not B is B a contradiction in terms. But it 
evidently implies that there can be no As which 
are not Bs, and thus we return to every A 
is B. 



xxi.] QUESTIONS AND ANSWERS. 203 

The above is a transcript with altered symbols from 
De Morgan s sixth example (Formal Logic, p. 123). But 
the contradiction arises simply from an error in not multi 
plying both alternatives by b. De Morgan follows the rule 
for the resolution of dilemmas, not observing that this 
rule can apply only when the alternatives are different. 
Equationally we have 

A = AB -I- AB. 

hence . . . Kb = AB -|- AB = o 
he gets . . . A3 = AB -|- AB - AB. 



It is rarely we find De Morgan tripping. 

33. Every A is one only of the two B or 
C ; D is both B and C, except when B is 
E, and then it is neither ; therefore no A 
is D. 

This problem was proposed by De Morgan in the Formal 
Logic, p. 124, and a solution has been given in the Principles 
of Science, p. 101 (first ed., vol. i. p. 117). The premises, 
as Professor Groom Robertson has pointed out, may be 
stated in two propositions, namely 

A = AEc -I- AC. 
D = D<?BC -I- DEfo 

Some objection has, however, been taken by Mr. Monro 
to my solution, and the student will find a good exercise 
in going over the solution carefully. It seems rather doubt 
ful how we should treat the combinations which are E 
and not B ; but the difficulties lie wholly in the interpreta 
tion of De Morgan s conditions. 



204 EQUATIONAL LOGIC. [CHAP. 

34. From A follows B, and from C follows D ; 
but B and D are inconsistent with each other. 
Hence A and C are inconsistent with each 
other. 

This problem which is formally the same as one of 
De Morgan s (Formal Logic, p. 123, Example 3), has the 
conditions 

(1) A = AB; 

(2) C-CD; 

(3) B = B*/. 

The consistent combinations are 



abCD. 

abcD. 
abed. 

We see that A and C never occur together, and in fact 
that A is never found excepting in the presence of B and 
the absence of both C and D. 

I committed an error in treating this problem in the 
Substitution of Similars (pp. 52, 53), by regarding a&cd 
as negatived by the premises. B may occur in the absence 
of A, but C and D must both be absent. 

35. What are the combinations of the qualities 
A, B and C which are possible according to 
the following conditions? (i) Where A is 
present, B and C are either both present at 
once or absent at once ; (2) where C is present, 
A is present. Describe the class not-B under 
these conditions. 



QUESTIONS AND ANSWERS. 205 

The conditions are expressed equationally as 

(1) A = ABC -I- Mf. 

(2) C = AC. 

The consistent combinations are shown in the margin ; 



ABC 



abc 



and A^C are removed by the first condition, 
and #BC and abC by the second. Selecting the 
two remaining ones which contain , we have the 
required discription 



b = hbc { abc = be. 
Where B is absent, C also will be absent. 

36. The logical value of two affirmative pre 
mises in the second figure is absolute zero. 
Examine the truth of this statement. [P.] 

The two premises assumed to be universal may be 
symbolised as 

(1) A = AB; 

(2) C - CB. 

The first negatives the combinations AC and Kbc, the 
second AC and abC, so that the premises overlap in 
regard to A^C. There remain five combinations. If we 
inquire what is A we get the value 

A = ABC ] KEc = AB, 



which is no more than (i ). For the description of C similarly 
we get (2). Thus it is plain that no relation is established 
between A and C. Concerning B we have even less 
information; for 

B = ABC -I- AB^ -I- rtBC -|- aEc = AB -|- aB = B. 



206 EQUATIONAL LOGIC. [CHAP. 

Of the negative terms, however, we draw more significant 
descriptions ; thus 

a = aEC -I- ac. 

b abc. 

c = EC -|- abc. 

It cannot be truly said that the logical value of the 
premises is absolute zero. 



37. Given that (i) everything which is B but not 
D is either both A and C or neither A nor C ; 
and (2) that neither C nor D is both A and B: 
prove that no A is B. 

[Adapted from Moral Science Tripos, 
Cambridge, 1879.] 

The conditions are 

(1) B^=B^(AC -I- ac). 

(2) C -I- D = (C ! D) (a -I- b). 

Confining our attention to the combinations containing 
AB, we see that ABcd is contradicted by (i), and the 
rest which contain either C or D, by (2). Hence there are 
no ABs, or no A is B. 

The equation (2) may be more briefly stated as 

AB = AB*/. 

The only combination containing a removed by (i) and 
(2) is 0BG 

38. Illustrate the use of symbolic methods by 
expressing the propositions 



xxi.] QUESTIONS AND ANSWERS. 207 

(1) No A is B except what is both C and 

D, and only some of that. 

(2) Either C or D is never absent except 

where A or B is present, but both are 
always absent then. [C.] 

The first proposition appears to deny the presence of any 
combination containing AB except there be also present C 
and D, and only in some cases then. To express this some 
we must introduce another letter term, say E, so that where 
E is present the above holds true ; where E is absent, A is 
not B at all. We find then that the following combinations 
are negatived : 

ABG/E. ABCD^. 

ABdDE. ABGfc. 

ABa/E. AEcDe. 



All this may be expressed in the one equation 
AB = ABCDE. 

The proposition (2) is not easy to interpret, but seems to 
mean 

A -I- B = cd. 



39. Every X is either P, Q, or R ; but every P 
is M, every Q is M, every R is M ; therefore 
every X is M. 

De Morgan, who gives the above (Formal Logic, p. 123, 



2 o8 EQUATIONAL LOGIC. [CHAP. 

Example 5), describes it as a common form of the dilemma. 
It is thus solved equationally : 

(1) X = X(P -I- Q -I- R); 

(2) P = PM; 

(3) Q= QM; 

(4) R = RM. 

Substituting by (2) (3) (4) in (i), 

X = X(PM -I- QM -I- RM); 
X = X (P -I- Q -I- R) M. 

Re-substituting in the last by (i) 
X - XM. 



40. Every A is either B, C, or D ; no B is A ; 
no C is A ; therefore, every A is D. 

[De Morgan, Formal Logic, p. 122.] 

The premises are clearly 

(1) A = AB -I- AC -I- AD. 

(2) B = 0B. 

(3) C = C. 

In (i) substitute the values of B and C given in (2) and 
(3), and then strike out two self-contradictory terms 

A == AflB ! A0C ! AD - AD. 



41. If A be B, E is F ; and if C be D, E is F ; 
but either A is B, or C is D ; therefore, E is 
F. (De Morgan, Formal Logic, p. 123.) 



xxi.] QUESTIONS AND ANSWERS. 209 

This appears to be more complicated in symbols than it 
really is. The first two premises are 

(1) AB = ABEF 

(2) CD - CDEF. 

To express the third premise we must introduce explicitly 
the tacit term, say X, meaning the circumstances under 
which the proposition holds good, in this place, or at 
this time, or under certain assumed conditions. Thus we 
have 

(3) X - XAB -I- XCD. 

substituting by means of (i) and (2), 

X - (XAB -I- XCD)EF, 
and re-substituting by (3) 

X = XEF. 

42. Every A is either B or C, and every C is A. 
This, says De Morgan (ibid. p. 123), is wholly 
inconclusive, and leads to an identical result. 

Equationally treated this is not quite so. The premises 
are 

(i) A - AB -I- AC 
(a) C = AC 
Hence (3) A - AB -|- C. 

De Morgan finds that Kb is C, which C being A gives 
A is A a necessary proposition or truism. But we also 
get, multiplying each side of (3) by b, 

Kb = AB -I- bC = bC. 



In the absence, then, of B, there is identity between A 
and C, but in the presence of B, A may be either B or C. 

P 



210 EQUATIONAL LOGIC. [CHAP. 

43. Every A is B or C or D ; every B is E ; every 
C is E ; and every E is D. 

[De Morgan, ibid. p. 123, Example 4.] 

Thus symbolised 

(i) A - AB -I- AC -I- AD. 
(2) B = BE. (3) C = CE. (4) E = ED. 

By obvious substitutions, by (2) and (3) in (i), and then 
by (4) in the result, we get 

A ="ABDE -I ACDE ! AD. 

But the first two of these alternatives are superfluous ; 
they both involve D and are therefore contained in the wider 
term AD. Hence 

A = AD. 

44. * If the relations A and B combine into C, it 
is clear that A without C following means that 
there is not B, and that B without C following 
means that there is not A. 

[De Morgan, Third Memoir on the Syllogism, p. 48.] 

The relations A and B combining into C appears to mean 
simply that AB is accompanied by C, or 

AB = ABC. 
To find A without C following, we have necessarily 

Kc - ABr -I- Afo 
Inserting for AB in this last its value ABC. 

Kc = ABG: i Kbc = Kbc. 
Similarly for B without C following 

B<: = AB<: -I- d&c = ABG: I c&c = d&c. 



xxi.] QUESTIONS AND ANSWERS. 211 

45. Suppose a class S to be divided (i-) on one 
principle into A and B, and on another prin 
ciple (2) into C and D, the divisions being 
exhaustive ; suppose further that (3) all A is 
C, and (4) all B is D ; can you conclude that 
all C is A, and all D is B ? [E.] 

The meaning of this problem appears to be that the class 
S will, as regards A and B, consist of SA$ and S#B, and 
similarly as regards C and D ; if so, there will under the 
first two conditions be only four possible combinations, 
namely 

SAJCtf. SaBG/. 

SAM). SaBd). 

But the further condition (3) negatives SA&rD, and (4) 
negatives S#BC</, so that, on inquiring for the description 
of C, we find it is (within the class S), A.bCd- } similarly D 
is a~BcD, Both questions then may be answered in the 
affirmative, provided that we are not to look beyond the 
sphere of the class S. 

46. What are the classes of objects regarded as 
possessing or not possessing the qualities 
A, B, C, D, which may exist consistently 
with the fundamental Laws of Thought, and 
the conditions that no class possesses both 
A and B, and that everything which does not 
possess B possesses C but not D ? [L.] 

The first condition that no class possesses both A and B 
will be sufficiently expressed in the premise A = A, which 



212 



EQUATIONAL LOGIC. 



[CHAP. 



prevents A and B from meeting. The second condition is 
obviously b = bCd. On going over the sixteen combina 
tions in the fifth column of the Logical Alphabet (p. 181), 
it will be obvious that the first four, containing AB, are 
negatived by the first premise. The third four 
(#B) remain untouched : of the second and fourth 
f ours con taining , all are negatived except KbCd 
an( * a ^d> The adjoining list of combinations is 
abCd therefore the answer to the question. 



h.bCd 
BCD 
aBCd 



AB 



47. How can we represent analytically the precise 
meaning of the opposition between a universal 
affirmative proposition and its contradictory, 
say between All As are Bs, and some As are 
not Bs ? 

The universal affirmative is symbolised as A = AB, and 
its logical power is to negative the combination A, as 
shown in the margin. Now some As are Bs 
was before explained to mean one A at least, it 
may be more or all As. But, even if there be 
one Ab found, it establishes the existence of the 
combination, subject to remarks elsewhere made 
(p. 142). In this qualitative treatment of logic number 
enters not at all, so that one counts for as much as a 
million. The force of the particular negative proposition is, 
then, to restore the combination which had been removed 
by the universal affirmative. 

48. If to the premises of an affirmative sorites we 
add a proposition affirming the first subject of 



ab 



xxi.] QUESTIONS AND ANSWERS. 213 

the last predicate, the conditions now become 
equivalent to an equally numerous series of 
identities, or doubly universal propositions in 
Thomsons 5 form U. 

Symbolically, if we have the series of premises A = AB, 
B = BC, C = CD, and so on, up to X = XY, and we then 
add the condition Y = AY, the premises immediately 
become the same in logical force as 

A = B = C = D=.... = X = Y. 

To give a perfect demonstration of this theorem might 
not be very easy ; but the student may convince himself of 
its truth by observing in several trials that the combinations 
consistent with the premises of a sorites as shown above, 
never contain a negative letter to the right hand of a positive 
one in the usual order of the alphabet. Thus the com 
binations consistent with the first two premises are 

ABC, aBCy abC and abc\ 
those for the first three are 

ABCD, rtBCD, 00CD, abcD and abed. 

Hence the last predicate appears in every combination 
except the last, and the first subject only in the first 
combination. In affirming the first subject of the last 
predicate, then, all combinations except the first, which 
contains both terms, and the last, which contains neither, 
must disappear. There remain in every case only the two 
combinations ABCDE .... XY .... and abcde .... 
xy . . . . which proceed from the identities stated in the 
question. 



2i 4 EQUATIONAL LOGIC. [CHAP. 

Suppose a pillar of circular section to be so shaped that 
no lower section is of less diameter than any upper section, 
but the section at the bottom is not greater than the section 
at the top ; we have here a physical analogue to the heap 
of propositions described above. 

49. Is Professor Alexander Bain correct in the 
following extract from his Deductive Logic 
(P- 159)? 

( T ) Socrates was the master of Plato. 

(2) Socrates fought at Delium. 

(3) The master of Plato fought at Delium. 

* It may fairly be doubted ^whether the transitions, in this 
instance, are anything more than equivalent forms. For the 
proposition (4) " Socrates was the master of Plato, and 
fought at Delium," compounded out of the two premises, is 
obviously nothing more than a grammatical abbreviation. 
No one can say that there is here any change of meaning, 
or anything beyond a verbal modification of the original 
form. 

Professor Bain in writing the above was clearly in need 
of means of more accurate analysis than his logical studies 
had afforded him. For if we put 

A = Socrates ; B = master of Plato ; 

C = one who fought at Delium, 

the premises are certainly 

(1) A = B; 

(2) A = AC. 

The conclusion (3) as it stands is B = BC, which negatives 
only two combinations AB<: and aBc, whereas the premises 



xxi.] QUESTIONS AND ANSWERS. 215 

negative in addition the three A^C, Afc, aRC. It is possible, 
indeed, to draw the conclusion (5)6 = AC, which is better 
than (3) by two combinations, namely, A&C and BC. As 
to the supposed proposition (4), it cannot be made into any 
non-disjunctive proposition without a change of meaning ; 
for, whether we make it into (6) A = BC, or A = ABC, it 
differs in force from the premises (i) and (2), which propo 
sitions in fact cannot be condensed into any single non-disjunc 
tive proposition of equivalent meaning. The fact is that the 
supposed proposition (4) consists merely of the two (i) and 
(2) re-stated in one compound sentence. It is not the 
proposition at all; it is the propositions. The case 
would have been considerably altered, indeed, had Mr. 
Bain interpreted (i) as Socrates was a master of Plato, of 
the form A = AB. The type of the premises would then 
have been essentially altered; but that he does not so inter 
pret it is obvious from (3), in which we have the master, 
not a master. Altogether this page of Professor Bain s 
work affords remarkable evidence of the inability of a most 
acute logician to maintain accuracy of logical vision without 
the aid of some kind of calculus like that developed in the 
latter part of this work. 

I append logical diagrams which almost explain them 
selves, the combinations pointed out by each bracket being 
those negatived by the proposition whose number is attached 
to the bracket. 

ABC 



(2) : , 
"U- 




(3) 



abC 

abc 




216 EQUATIONAL LOGIC. [CHAP. 

50. How far does the conclusion of an Aristo 
telian syllogism fall short of giving all the 
information contained in the premises ? 

The premises of Barbara, say A = AB, B = BC, nega 
tive four combinations, AB<r, A$C, A&*, a&c. The conclusion 
A = AC negatives only two of these, namely, ABc and 
AJbc. Measured in this way, then, it contains only half of 
the information of the premises; but of course if the 
conclusion gives just that information which is desired, the 
overlooking of the rest is no harm. Enough is as good as 
a feast or rather better. 



51. Take the premises of a syllogism in Barbara, 
such as (i) all As are Bs, and (2) all Bs are 
Cs, and determine precisely what they affirm, 
what they deny, and what they leave in 
doubt. 



ABC 



\~Kbc 
aQC 



To answer this question, we must form 
the eight combinations of A, B, C and 
their negatives, as in the margin; we 
then strike out AC and hbc as being in 
conflict with condition (i), and AB<r and 
a~Bc as being similarly in conflict with the 
condition (2), that all Bs are Cs. There 
remain four combinations, ABC, #BC, abC, and abc. But 
these do not stand on the same logical footing, because if 
we were to remove ABC, there would be no such thing as 
A left ; and if we were to remove abc there would be no 
such thing as c left. Now it is the Criterion or condition of 
logical consistency (p. 181) that every separate term and its 



abC 
abc 



xxi.] QUESTIONS AND ANSWERS. 217 

negative shall remain. Hence there must exist some things 
which are described by ABC, and other things described by 
abc. But as regards the remaining two combinations, #BC 
and abC, the case is different; for we may remove either, or 
both of these without wholly removing any term. We 
might add to the premises the new condition that all BCs 
are As, or BC = ABC, which would negative #BC ; or we 
might add the condition, all Cs are As, or C = AC, which 
would remove both 0BC and abC. 

We may sum up the meaning of the original premises (i) 
and (2) by saying that they deny the existence of AB<r, 
AC, hbc, and aEc ; that they affirm the presence or logical 
existence of ABC and abc; and thirdly, while leaving #BC 
and abC uncontradicted, they are consistent with the presence 
or absence of these two combinations. This is all that they 
leave in doubt concerning the relations of A, B, and C. 

52. What is the amount of contradiction in the 
following celebrated epigram ? 

The Germans in Greek, 
Are sadly to seek ; 

* * * * 

All, save only Hermann, 
And Hermann s a German. 

Putting A = German ; B = Hermann ; 

C = sadly to seek in Greek, 

the premises are evidently 

(1) A = AC. 

(2) B = B* 

(3) -B = AB. 



218 EQUATIONAL LOGIC. [CHAP. 

The logical diagram is as in the margin; it will be 
noticed that B disappears entirely, indi- 



eating contradiction ; but A remains in 
the combination AC, It is obvious that 



the wit of the epigram arises from the per 
ception of contradiction. (See Hamilton s 
Lectures , vol. iii. p. 393.) 



ABC 



BC 



(2) 



abC 

abc 



53. Show that you can make no assertion about 
two terms A and B (and these only), which 
is not either contained in the assertion of 
identity (A = B), or else contradictory thereto. 

The proposition A = B removes two out of the four 
combinations thus 

Consistent Inconsistent 

Combinations. Combinations. 

AB. A. 

ab. tfB. 

Now, if any new assertion negatives either or both of Kb 
and flB, it must be an assertion contained in and inferrible 
from A = B. If it removes either AB or ab, it must con 
tradict A = B, because either A and B or a and b will then 
disappear entirely from the Logical Alphabet. It might be 
said perhaps that a new assertion could remove one consis 
tent and one inconsistent combination, for instance, ab and 
Ad; but this cannot be done except by a contradictory 
assertion. Any other pairs such as AB and Kb, AB and 
0B, or ab and a~B, being removed, removes some letter 
entirely and involves contradiction. 



xxi.] QUESTIONS AND ANSWERS. 219 

54. Is it (i) logically (2) physically possible that 
all material things are subject to the law of 
gravity, and that at the same time all not 
material things should be subject to the same 
law ? [L.] 

It is logically possible, that is to say, in accordance with 
the Laws of Thought, that all things material and all things 
not material should be subject to the law of gravity. In 
this case what is not subject to the law of gravity would be 
found among not-things. But it is not logically possible 
that all (material things) and all not-(material things) should 
be subject to the law of gravity, because this is equivalent 
to denying the existence of any class not subject to the law 
of gravity. This class would by one condition be not- 
material, and by the other condition it would be material, 
which is impossible. But by the law already described 
(p. 181) as the Law of Infinity, every logical term must 
be assumed to have its negative. The student is recom 
mended to work out this question with the aid of letter 
symbols. 

As to the second part of the question, what is not logically 
possible is of course not physically possible. Hence we are 
restricted to the inquiry whether it is physically possible that 
all things material and all things not-material should be 
subject to the law of gravity. This can only be answered 
on logical grounds thus far, that if the property of gravita 
tion is essential to material things and forms a part of the 
definition of them, then it is not possible that not-material 
things should gravitate. As a matter of fact the possession 
of inertia is perhaps the ultimate test of materiality ; but 
gravity is proportional to inertia and is an equally good 
test. 



220 EQUATIONAL LOGIC. [CHAP. 

55. It is observed that the phenomena A, B, C 
occur only in the combinations ABc, abC, and 
dbc. What propositions will express the laws 
of relation between these phenomena ? 

Of the eight combinations of A, B, C, only these three 
remain. As we see that A occurs with and only with B, and 
a with and only with #, it is firstly obvious that A = B is 
the chief law. But as this law of relation leaves the com 
bination ABC uncontradicted, we must have a second law 
to remove this, which may be either AB = AB<r, or else 
B = B<r. Observe, however, that the laws A = B and 
B = B<: overlap and are pleonastic, because they both 
deny that B can be #BC. Hence the simplest statement 
of the laws of relation is 

A - B. 
AB - ABr. 

56. Given three terms, for instance, water, blue, 
and fluid, how would you proceed to ascertain 
the utmost number of purely logical relations 
which can exist among them ? [L.] 

The relations of any three terms or things or classes of 
things must be governed in the first place by the universal 
Laws of Thought (p. 180). These laws restrict the combi 
nations of three things, present or absent, to eight at the 
utmost; for each thing may be present or absent giving 
2x2x2 = 8 cases. But any special logical relation 
which may exist between the things has the effect of 
further restricting these combinations ; the relation that 
water is a fluid, prevents the existence of the combination 



xxi.] QUESTIONS AND ANSWERS. 221 

water, not-fluid. Conversely the removal from the series of 
any one or more of the eight combinations expresses the 
existence of a relation or relations negativing the existence 
of these combinations. Thus, the removal of the two 
combinations water, not-blue, fluid ; water, not-blue, not- 
fluid, expresses the law that all water is blue. Thus the 
logical meaning of any condition is represented by the state 
of the combinations agreeing with those conditions. It 
follows that the utmost possible number of distinct logical 
relations will be ascertained by taking the eight possible 
combinations of the three terms and striking out one or 
more of the combinations in every possible variety of ways. 
The number of these ways cannot exceed 256 ; for each of 
the eight combinations may be either present or absent, 
giving 2x2x2 x 2 X2 x 2 x 2x 2 = 256 ways. 
But this calculation will include many cases where one or 
more of the three terms and their negatives disappear alto 
gether, representing contradiction in the conditions. Many 
different selections, too, proceed from logical relations 
similar in character and form ; thus the law A = AB is 
similar to A = A, and to a ab ; the law A = BC -|- be 
is similar to C = AB ) ab ; and so forth. The investiga 
tion is fully described in the Principles of Science (pp. 134- 
143 j ist ed. vol. i. pp. 154-164) as also in the Memoirs of 
the Manchester Literary and Philosophical Society^ Third 
Series, vol. v. pp. 119-130. It is found that the 256 
possible selections are thus accounted for 

Proceeding from consistent logical conditions 192 
inconsistent 63 

no condition at all i 



256 



222 



EQUATIONAL LOGIC. 



[CHAP. xxi. 



The consistent logical conditions are found, however, on 
careful analysis to fall into no more than fifteen distinct 
forms, or types of relation, which are stated in the following 
table 



Reference 
Number. 


Propositions expressing the 
general type of the logical 
conditions. 


Number of dis 
tinct logical 
variations. 


Number of 
combinations 
contradicted 
by each. 


I. 


A = B 


6 


4 


II. 


A = AB 


12 


2 


III. 


A = B, B = C 


4 


6 


IV. 


A = B, B = BC 


24 


5 


V. 


A = AB, B = BC 


24 


4 


VI. A = BC" 


2 4 


4 


VII. j A = ABC 


2 4 


3 


VIII. ! AB = ABC 


8 


i 


IX. ! A - AB, B = aEc 


2 4 


3 


X. A = ABC, at = abC 


8 


4 


XL 


AB = ABC, al> = abc 


4 


2 


XII. AB = AC 


12 


2 


XIII. 


A = BC ! Kbc 


8 


3 


XIV. 


A - BC ! be 


2 


4 


XV. 


A = ABC, a = aEc -\- abC 


8 


5 








i 



CHAPTER XXII. 

ON THE RELATIONS OF PROPOSITIONS INVOLVING THREE OR 
MORE TERMS. 

i. THE doctrine of the opposition of propositions, exhi 
bited in the well-known square, is an important and inter 
esting fragment of ancient logic; but it is now apparent 
that propositions involving only two terms, one subject and 
one predicate, do not sufficiently open up the question of 
the relationship of propositions. Two terms admit of only 
four combinations, and these can be present and absent 
only in sixteen ways, nine of which involve contradiction. 
There remain only seven cases of logical relation which 
resolve themselves into only two distinct types of propo 
sition. (Principles of Science, pp. 134-7; ist ed. vol. i. 
p. 154-7.) With the introduction of a third term the 
sphere of inquiry becomes immensely extended. There 
are now, as we have seen (p. 221) 193 different cases of 
selection of combinations resolving themselves into fifteen 
distinct types of relation. The possible modes of relation 
of one proposition to another, including under the expres 
sion one proposition any group of propositions, become 
considerably complex. Such modes of relation seem to 
be seven in number : thus one proposition is as regards 
another 



224 EQUATIONAL LOGIC. [CHAP. 

(1) Equivalent. 

(2) Inferrible, or contained in the other, but not equi 

valent. 

(3) Partially inferrible and otherwise consistent. 

(4) Consistent but indifferent and not inferrible. 

(5) Partially inferrible, partially contradictory. 

(6) Partially indifferent, partially contradictory. 

(7) Contradictory. 

2. Let us take as an example the proposition 
Steam = aqueous vapour, 

and give a pretty complete analysis of its related pro 
positions. 

Let A = steam ; 

B = aqueous ; 

C = vapour. 

The proposition being evidently of the form 

(i) A = BC, 

the combinations contradicted will be as in the margin. 
ABC The equivalent proposition will be 

Not steam = not aqueous or not vapour. 
An inferrible but not equivalent assertion 
will be any one which negatives one, two, 
or three, but not four of the combinations 
negatived by (i). There will therefore be 

4x3 4x^x2 * * 

4 + - - + or 1 4 such infer- 

1x2 1x2x3 

rible and logically distinct propositions. We may infer 
steam is aqueous ; steam is vapour ; what is not vapour 
is not steam; what is not aqueous is not steam; non- 
aqueous vapour is not steam ; and so forth. 




XXIL] RELATIONS OF PROPOSITIONS. 225 

The third class of related propositions will include those 
which negative one or more of the excluded combinations, 
and one or more indifferent combinations. Indifferent 
combinations, as the name expresses, are those which can 
be removed without wholly removing any of the letters 
A, B, C, a, b, c. In this case any one of the remaining 
combinations except ABC may be singly removed. Thus 
not steam is not aqueous, or in letters a a b, is not 
contradictory to (i) and it may be inferred from (i) in 
respect of vapour which is not steam. But the assertion 
that other things which are not steam are not aqueous is 
not inferrible, but is consistent with (i). A proposition, 
again, which should negative A^C, Kbc, aRc, abC will be 
inferrible in respect of the two former, and consistent in 
respect of the two latter combinations. To ascertain what 
such proposition is we must look in the Logical Index, 
afterwards described, for the proposition which leaves a fi e 0, 
and we find in the 55th place b ac, or not-aqueous = 
not-steam and not-vapour. The other possible propositions 
of the same class are numerous and various. 

To obtain one of the fourth class, which is merely con 
sistent and indifferent, we must take any one or more of 
the combinations unnegatived by (i), for instance a&C, in 
such a way as not wholly to remove any letter. Thus 
ab = abc, or not-steam which is not aqueous is not vapour 
is an assertion quite indifferent to (i). So is the assertion 
#B = #C (Logical Index, No. 7). 

Contradictory propositions being defined as those which 
wholly remove any term, such will be any one which re 
moves ABC. Thus to say that steam is not aqueous is a 
case of the 5th class; it is inferrible from (r) in respect of 
steam which is not vapour (AB^), but it is contradictory 
because it also negatives steam which is vapour. 

Q 



226 EQUATIONAL LOGIC. [CHAP. xxn. 

A proposition of the sixth class is discovered by taking 
any combination which may be spared with one which 
cannot, such as abC and ABC, and looking in the index, 
we find AC - ^C, or steam- vapour is identical with non- 
aqueous vapour, as a partially consistent, partially contra 
dictory proposition as regards (i). It may or may not be 
true that what is not steam and not aqueous is not vapour, 
but it is contradictory to (i) to say that vaporous steam is 
not aqueous. 

An example of a simply contradictory proposition of 
Class 7 is found in one which removes ABC only, such as 
AB = ABr ; again a = aE, or not-steam is aqueous deletes 
b ; c = AEc deletes c. 

2. As a second example, let us take the propositions 
(i) Hand = right-hand or left-hand; (2) Right is not left. 

Putting A = hand; B = right; C = left; the conditions 
are evidently 

(1) A = AB -I- AC. 

(2) B = EC. 

The consistent combinations are shown in the margin, 
ABC and the student ma y verify the following 
list, which gives one specimen of each of 
(2) the seven classes of related assertions, the 
reference number of the Logical Index 
being also added. 

(1) Equivalent. B = EC; be = abc. No. 153. 

(2) Inferrible, etc. aE = aEc. No. 9. 

(3) Partially inferrible, etc. a = abc. No. 15. 

(4) Consistent, etc. AB - ABC; ab = abc. No. 67. 

(5) Partially inferrible, etc. C = AC; A = AB. No. 59. 

(6) Partially indifferent, etc. A - AB^; ab = abc. No. 179. 

(7) Contradictory, b = be. No. 35. 



dnC 

abC 

abc 



CHAPTER XXIII. 

EXERCISES IN EQUATIONAL LOGIC. 

I NOW give a small collection of examples and problems 
designed to enable the student to acquire a complete com 
mand of the equational and combinational views of logic. 
They are for the most part devised specially for this book, 
but a few have been utilised in examination papers, and a 
few have been adopted as indicated from the papers of other 
examiners. These questions form perhaps a partial answer 
to Professor Sylvester s remark, as quoted in the preface, 
especially when we observe that the questions and problems 
involving the relations of three terms can be multiplied 
almost ad tnfinttum, without resorting to like questions in 
volving four, five, or more terms. The student will readily 
gather that the number, variety, and complexity of problems 
in pure logic is simply infinite, and is such as we gain no 
glimpse, of in the old Aristotelian text-books. 

i. Represent equationally the following assertions : 

(1) With the exception of porcelain there is no non- 

metallic substance which has been employed to 
make coins. 

(2) With the exception of zinc and the metals discovered 

during the last hundred years, there is no metal 
which has not been employed to make coins. 



228 EQUATIONAL LOGIC. [CHAP. 

(3) The worth of that is that which it contains, 

And that is this, and this with thee remains. 

[SHAKESPEARE.] 

(4) It is dangerous to let a man know how far he is but 

a brute, without showing him also his grandeur. 
It is dangerous again to let him see his grandeur, 
without his baseness. It is [even more] dangerous 
to leave him ignorant in both ways ; but it is a 
high advantage to represent to him both the one 
and the other. (Pascal, Pensees.) 

2. Represent in the forms of equational logic any of the 
following arguments : 

(1) Milton was a great poet, and a fearless opponent of 

injustice ; a great poet should be honoured ; a 
fearless opponent of injustice should be honoured : 
therefore Milton should be honoured. 

(2) The virtues are either passions, faculties, or habits: 

they are not passions, for passions do not depend 
on previous determination ; nor are they faculties, 
for we possess faculties by nature ; therefore they 
are habits. 

(3) There can be no person really fit to exercise absolute 

power, because the qualifications requisite to fit a 
person for such a position would consist in native 
talent combined with early training ; now such a 
talent cannot be possessed in early childhood. 
(Suggested by De Morgan, Syllabus, p. 67.) 

(4) One of the masters of chemistry was Berzelius ; 

Berzelius was a Swede ; 

One of the masters of chemistry was a Swede. [D.] 

(5) This heavenly body is either a planet or a fixed star ; 



xxiii.] EXERCISES. 229 

all fixed stars twinkle; planets do not twinkle; 
this body twinkles, therefore it is a fixed star. 
(6) Show me any number of men, and I will say with 
confidence, either that they will with one accord 
raise their voices for liberty, or that there are 
aliens among them. (The stump orator s mode, 
according to De Morgan, of saying that all 
Englishmen are lovers of liberty.) [B.] 

3. Infer all that you possibly can, by way of contra 
position or otherwise, from the assertion all A that is 
neither B nor C is D. [R.] 

4. Express equationally Miscellaneous Example No. 39 
in Elementary Lessons in Logic ; p. 317. 

5. What proposition concerning nebulas and vaporous 
bodies leaves doubtful the existence of a class of things 
which are neither nebulae nor vaporous bodies ? 

6. Represent the fact that A differs from B in two equiva 
lent equational propositions. 

7. Prove equationally that the proposition, All elements 
are either metal-elements or elements, is a mere truism. 

8. What is the difference between the propositions 
A = AB -|. A, B = AB .|. B, and A = B .|. A? 

9. Prove that if all not-Bs are not-As, and all Bs are As, 
then A = B, and vice versti. 

10. Show that the negative premises No As are Bs and 
no Cs are Bs, imply the logical existence of a class B which 
is neither A nor C. 

1 1. Prove the equivalence of the following assertions : 

(1) Every gem is either rich or rare. 

(2) Every gem which is not rich is rare. 

(3) Every gem which is not rare is rich. 

(4) Everything which is neither rich nor rare is not a gem. 



230 EOUATIONAL LOGIC. [CHAP. 

12. Show that if metals which are either not valuable or 
not destructible are unfitted for use as money, it follows 
that destructible metals which are fitted for use as money 
must be valuable. 

13. Does the proposition A = B -|- BC differ in force 
from A = B ? 

1 4. All animals having red blood corpuscles are identical 
with those having a brain in connection with a spinal cord. 
What is the description you may draw from this proposition 
of things having a brain not in connection with a spinal 
cord ? 

15. Luminous body is either self-luminous or luminous 
by reflection ; melted gold is both self-luminous and lumi 
nous by reflection. Unmelted gold is not self-luminous 
but is luminous by reflection. Represent these premises 
symbolically, and draw descriptions of the terms (i) lumi 
nous body, (2) self-luminous body, (3) body luminous by 
reflection, (4) body not luminous, (5) body not self-luminous, 
(6) not melted gold, (7) not unmelted gold. 

1 6. There are no organic beings which are devoid of 
carbon. Determine precisely what this proposition affirms, 
what it denies, and what it leaves doubtful. 

17. Prove the equivalence of the following statements 
No right-angled triangles are equilateral; no equilateral 
triangles are right-angled ; no right-angled equilateral figures 
are triangles. 

1 8. All scalene triangles have their three angles equal to 
two right angles. What are the least or simplest assertions 
which added to the above will make it equivalent to All 
triangles are all figures which have their three angles equal 
to two right angles ? 



xxiii.] EXERCISES. 231 

19. All equal -sided squares have four right angles. 
What is the least extensive proposition which added to the 
above makes it equivalent to All squares are equal-sided 
and have four right angles ? 

20. If an orator were to assert that Afghanistan is a very 
poor country, but it is essential to the security of India, but 
a reporter were to consolidate these two assertions into the 
one assertion that a very poor country, Afghanistan, is the 
Afghanistan which is essential to the security of India, how 
far would the reporter have misrepresented the logical 
meaning of the orator ? 

21. Express the following argument equationally : Every 
organ of sense has nervous communication with the brain ; 
for such is the case with all the five organs of sense, the 
eye, ear, nose, tongue, and skin. 

22. If requested to draw from the assertion All coal 
contains carbon a description of the term metal, what 
answer would you give ? 

23. What values will you obtain for the terms man, brute, 
and gorilla, under the conditions that a gorilla is a man, and 
that all men are included and all gorillas excluded from the 
class of non-brutes ? 

22. Assuming that armed steam-vessels consist of the 
armed vessels of the Mediterranean and the steam-vessels 
not of the Mediterranean, inquire whether we can thence 
infer the following results : 

(1) There are no armed vessels except steam- vessels in 

the Mediterranean. 

(2) All unarmed steam-vessels are in the Mediterranean. 

(3) All steam-vessels not of the Mediterranean are armed. 



232 EOUATIONAL LOGIC. [CHAP. 

(4) The vessels of the Mediterranean consist of all 
unarmed steam-vessels, any number of armed 
steam-vessels, and any number of unarmed vessels 
without steam. (Boole, The Calculus of Logic, 
Cambridge and Dublin Mathematical Journal^ 
1848, vol. iii. pp. 1991.) 

25. How would you otherwise describe the class of things 
which are excluded from the class of white, malleable, 
metals ? 

26. Show that the description of the class of things 
which are not (either A, or if not A then both B and C), is 
as follows either not-A and not-C, or if it be C then both 
not-A and not-B. 

27. How do any two of the three equations A = B 
B = C, C = A, differ in logical force from the third ? 

28. Frame a sorites with one premise negative and one 
particular, and represent it equationally. 

29. Contrast the logical force of each of the proposi 
tions A = AB -I- AC -I- AD .|. . . and A = ABCD . ., 
with that of the group of propositions A = AB, A = AC, 
A = AD, etc. ; point out, moreover, which can be inferred 
from which. 

30. Show that, under the condition of our Criterion of 
Logical Consistency (p. 181) the assertion that there are no 
such things as fresh-water foraminifera, involves the asser 
tion that there are foraminifera which are not fresh-water 
beings, and fresh-water beings which are not foraminifera, 
but leaves doubtful the occurrence of things which are 
neither fresh-water beings nor foraminifera. 

31. From the premises, All gasteropods are mollusca, 
and no mollusca are vertebrates, obtain descriptions of 
the classes gasteropods and invertebrates. 



xxm.] EXERCISES. 233 

32. Eloquence should contain both what is agreeable, 
and what is real ; but what is agreeable should be real 
(Pascal, Pensees). Represent the above symbolically, 
putting A = component of eloquent speech, B = agreeable, 
C = real. 

33. Assuming it to be known that all mammals have red 
blood corpuscles, and that they also have vertebrae, invent 
five or six other distinct assertions which you might make 
about mammals, the possession of red blood corpuscles, and 
the possession of vertebrae, including of course the negatives 
of these terms, without coming into logical conflict with the 
known relations of the terms as above stated. 

34. How would you otherwise describe the class of things 
which are excluded from the class of non-crystalline solids 
which are either non-metallic non-conductors, or else metal 
lic conductors, and which are moreover either brittle and 
in that case useless for telegraphy, or else malleable and in 
that case useful for telegraphy ? 

35. Compare the following propositions : 

(1) X is Y. 

(2) X is Y and is in some cases Z, and in some cases 

not Z. 

By the law of excluded middle we know that X must be 
either Z or not Z. Is then the sentence (i) precisely 
identical in logical force with (2)? Compare now the 
following definitions : 

(3) A right-angled triangle is that which has a right 

angle. 

(4) A right-angled triangle is that which has a right angle, 

and of which two sides are or are not equal. 
Are these definitions precisely identical in logical force ? 

[C-] 



234 EQUATIONAL LOGIC. [CHAP. 

36. What is the difference between saying that sea-water 
is drinkable and not scarce, and saying that drinkable sea- 
water is not scarce ? 

37. If from the premises All rectangles are parallelo 
grams, and Parallelograms consist of all four-sided figures 
whose opposite sides are parallel, we infer that all rectangles 
are parallelograms, being four-sided figures with opposite 
sides parallel, how far does this inference fall short of being 
equivalent to the premises ? 

38. To say that Adam Smith is the father of Political 
Economy and a Scotchman is as much as to say that he is 
a Scotch father of Political Economy, and that no one but 
he can be a father of the science. Give the symbolic proof 
of this equivalence. 

39. To lay down the condition that what is either A or 
else B, is what is both A and B or else both A and C and 
vice versd t is to state disjunctively what may be laid down 
in two non-disjunctive propositions asserting that A without 
B is C and also B must be A. 

40. Reduce the two assertions A = Kbc and a = ac to a 
single one. 

41. Give a good many inferences from the proposition 
A = B -|- AC, and also equivalents, distinguishing carefully 
between those inferences which are equivalent and those 
which are not. 

42. Develop symbolically the term Plant (A) with refer 
ence to the undermentioned terms (B, C, D, E, F), under 
the conditions that acotyledonous (b) plants are flowerless ; 
(c) monocotyledonous (D) plants are parallel-leaved (E); 
dicotyledonous (F) plants are not parallel-leaved ; and 



xxiii.] EXERCISES. 235 

every plant is either acotyledonous, monocotyledonous, or 
dicotyledonous, but one only of these alternatives. 

43. Completely classify triangles under the following 
conditions 

(1) Equilateral triangles are isosceles. 

(2) Scalene triangles are not isosceles. 

(3) Obtuse-angled triangles are not right-angled. 

(4) Acute-angled triangles have three acute angles. 

(5) Obtuse-angled triangles have not three acute angles. 

(6) Equilateral triangles are not right-angled. 

What other conditions must be added to comply with the 
results of geometrical science ? 

44. Among plane figures the circle is the only curve of 
equal curvature. Show that this is the same as to assert 
that a plane figure must either be a curve of equal curva 
ture, in which case it is also a circle, or else, not a circle 
and then not a curve of equal curvature. 

45. Which of the following propositions are equivalent to 
the first in the list ? 

(1) Crystallised carbon is not a conductor. 

(2) Carbon which conducts is not crystallised. 

(3) Conducting crystallised substance is not carbon. 

(4) Conductors are either not carbon or not crystallised 

substances. 

(5) Carbon is either not a conductor or not crystallised. 

(6) Conductors which are not carbon are crystallised. 

(7) Crystals are either non-conductors or not composed 

of carbon. 

(8) Crystallised conductors are carbon. 

46. Prove that any set of exclusive alternatives combined 
with part of that set produces only that part. 



236 EQUATIONAL LOGIC. [CHAP. 

47. Show that the conclusions of Celarent, Cesare, 
Camestres, and Camenes give in each case only half the 
information contained in the premises. 

48. Verify by various trials the statement that no inference 
by substitution within a group of propositions can negative 
combinations not negatived by the group of premises. 

49. Show that Cesare and Camestres belong to the same 
type of assertion as Barbara and Celarent. 

50. Assign the premises of the following moods of the 
Syllogism to their proper types of assertion : Darapti, 
Bramantip, Camenes. 

51. Prove that any proposition which is contradictory to 
common salt = sodium chloride, can be inferred, so far 
as it is contradictory, from the assertion common salt = 
what is not sodium chloride. 

52. Does it or does it not follow that any proposition of 
the ;;/th type (see pp. 221-2) will always be equally con 
tradictory to one of the wth type ? 

53. Refer to Boole s Laws of Thought, pp. 146149, 
and taking the premises of the complex problem there 
solved to be expressed in our system as follows : 

(1) ac = <wE(B// -I- D); 

(2) AD^ = AD^(BC -I- be); 

(3) A (B -I- E) = G/ [ cD ; 

work out the consistent combinations, and infer descriptions 
of the classes B, AC, AO>, D, e, AB, AB^, ab, AE, ACE, 
BD, DE, De, C, CD, etc. Verify by showing that D and e 
multiplied together give De and so forth. 

54. If Brown asserts that all metals are reputed elements, 
and that all reputed elements will be ultimately decomposed, 
whereas Robinson holds that all metals are reputed elements 



XXIIL] EXERCISES. 237 

which will be ultimately decomposed, what is the exact 
amount of logical difference between them ? 

55. Compare the logical force of all the following pro 
positions, and point out which pairs are equivalent, and 
which may be inferred from other ones. 

(1) A square is an equal-sided rectangle. 

(2) What is not equal-sided is not square. 

(3) What is not square is not equal-sided. 

(4) Equal-sided rectangles are squares. 

(5) No rectangle which is not equal-sided is square. 

(6) A square can be neither unequal-sided nor anything 

but a rectangle. 

(7) An unequal-sided square does not exist. 

56. Taking letters to represent qualities thus : A = 
having metallic lustre ; B = malleable ; C = heavier than 
water ; D = white coloured ; E = fusible with difficulty ; 
F = conducting electricity; form descriptions of each of 
the metals gold, silver, platinum, copper, iron, lead, tin, 
zinc, antimony, sodium, and potassium, and then exhibit 
the extension of the following classes : AB ; BC ; BCD ; 
BCF; A; be; B//; and so forth. 

57. Express symbolically the following classes of things 

(1) Hard, wet, black, round, heavy, stone. 

(2) Thing which is hard, wet, either black or red, but 

not round, and either heavy or not heavy. 

(3) Thing which is either not hard, or not wet, or not a 

stone, but is either black and then round, or heavy 
and then a stone. 

58. Referring to the Principles of Science (pp. 75 6 ; ist 
ed. vol. i. p. 90), develop all the alternatives of A as limited 
by the description 

A = AB {C -I- D (E -I- F)} 



238 EQUATIONAL LOGIC. [CHAP. 

and infer descriptions of the following terms, Ace, Acf, AB<rD. 
(See De Morgan, Formal Logic, p. 1 1 6 ; Third Memoir on 
the Syllogism, p. 12 in the Camb. Phil. Trans., vol. x.) 

59. Represent this argument symbolically: A straight 
line can cut a circle in two points, and similarly an ellipse, 
and a hyperbola ; but these are all the possible kinds of 
conic sections ; therefore a straight line can cut any conic 
section in two points. 

60. It being understood (i) that only the congenitally 
deaf are mute ; (2) that an uneducated deaf person is mute, 
but uses signs ; (3) that an educated deaf person is not 
mute, and does not use signs : express these conditions 
symbolically and describe the classes of persons who are 
deaf; mute; deaf-mutes; educated persons, etc. 

6 1. Show how by the process of substitution alone to 
sum up into one disjunctive proposition the assertion that 
John is mortal ; Thomas is mortal ; William is mortal. 

62. Prove that the premises of syllogisms in the moods 
Darapti and Felapton can be expressed in the form of a 
single non-disjunctive proposition, and assign its type. Show 
also that this is not the case with the moods of the other 
three figures. 

63. Prove that the following propositions or groups of 
propositions involve self-contradiction : 

(i) A = B -I- b. 

/ \ (B = AB 

^ i = A*. 

(3) A = AB ; B = BC ; C = aC. 

64. Analyse the force of Hamilton s form of proposition, 
* Some A is not some A, putting for some and some 
respectively the letter terms P and Q. 



EXERCISES. 239 

65. What does the assertion Some things are neither A 
nor B tell us about things which are not-A ? 

66. How far do the conclusions of the syllogisms in 
Darapti, Felapton, Bramantip, Camenes, and Fesapo, as 
deduced on p. 188, respectively fall short of containing all 
the information given in the premises ? 

67. Show that C = AC -|- BC is equivalent to the two 
propositions, AB = AB<: and ab = abc. Name the type. 

68. To say that whatever is devoid of the properties of 
A must have those either of B or of D, or else be devoid 
of those of C, is the same as to say that what is devoid of 
the properties of B and D, but possesses those of C, must 
have A. Prove this. 

69. What statement or statements must be added to the 
proposition, What is not a square is either not equal-sided 
or not a rectangle, in order to make the assertions in the 
whole equivalent to the definition of a square that it is an 
equal-sided rectangle ? 

70. What is the difference between the assertion A = 
ABC and the pair of assertions b ab, and c - be ? 

71. Prove that from one of the propositions, A = ABC, 
and AB = ABC, we can infer the other, but not vice 
versa, and point out which is the one which can be so 
inferred. 

72. Give three logical equivalents to the proposition, 
ACD = ACD. 

73. Demonstrate the equivalence of A = AB [ AC with 
Al> = AC, and with Ac = AB<*. 

74. Show how by substitution alone to obtain, A = AB 
from A = ABCD ; also obtain A = AC" and A = AD. 
(Principles of Science, p. 58 ; ist ed. vol. i. p. 69.) 



240 EQUATIONAL LOGIC. [CHAP. 

7 5. Verify the statement that any set of alternative terms 
combined with the same set, reproduces that set that is to 
say, show that AA = A when for A we substitute any one 
of the followin terms : 



ABC -I-0BC -I- abc\ 
AB^ ! AC -I- tfBC. 

76. Show by trial that if in any pair of logically equivalent 
assertions such as A = Kb and B = #B, we substitute 
for A and B any logical expressions, such as CD for A, 
and CE for B, and their negatives in like manner for the 
negatives of A and B, we always obtain new equivalent 
assertions. 

77. As a further example of equivalent assertions take 
the following pair of propositions : 

f AB = ABC, 
{ Ac = Me, 

and substitute as follows : 

A = PQ, B = Qr, C - PR. 

78. Express a - ab and hb - A^C in the form of a 
single disjunctive proposition. To what type does it 
belong ? 

79. Express equationally De Morgan s forms of propo 
sition (Formal Logic, p. 62). 

(1) Everything is either A or B ; 

(2) Some things are neither As nor Bs. 

80. Verify the identical equations 

A-l- B = A -I- #B; 

AB - A (a -I- B) 
A -I- BC = (A -I- B) (A -I- C). 



xxiii.] EXERCISES. 241 

8 1. Verify the following equivalences as transcribed from 
De Morgan s Syllabus, p. 42 : 

f A = (B ! C) D, J A = (B -I- C) (D -|- E), 

\ a be -|- </; ( = be -\- de ; 

f A = BC -|. D, j- A = B -I- C (D -|- E), 

\a = (H-^; U = (<M-<fc); 

J A = (B ! CD) (E ] FG), J A = B i C -|- JD, 

\a-bc -|- &/ i </!* <fj I = &* 

82. State all the propositions involving only the terms 
named which can be inferred from the equation, Stone = 
rock ; and all the propositions which are equivalent to this 
one, Stone = stone-rock. 

83. Show how by the mere process of substitution you 
can draw the proposition A = AD from the three propo 
sitions A = AB, B =- BC, and C = CD. 

84. What propositions added to A = AD are exactly 
equivalent in meaning to A = AB, B = BC, and C = 
CD jointly? 

85. If both A and B have the property C, but A never 
occurs where D is, and B never occurs where D is absent, 
what is your description of the class of things which are 
devoid of the property C ? 

86. The proposition A = A (B -|- C) being equivalent to 
b - ab -|- A^C, verify this truth by showing that it holds 
good when for A we substitute the term P^ -|-/Q, for B 
the term QRS, and for C the term q&s. 

87. If a person were, correctly or incorrectly, to define 
Members of Parliament (including Lords and Commons) as 
either peers not chosen by election, or else not-peers chosen 
by election, that is as much as to assert both that all members 
are non-elected peers and elected non-peers, as well as that 

R 



242 EOUATIONAL LOGIC. [CHAP. 

all who are not members comprise the two classes of persons 
who are neither peers nor elected persons, and those who 
being peers have been elected but cannot sit. 

88. It is not correct to say that because what is not A, 
but is B, is also C, therefore everything that is both B and 
C is A; but what further conditions may be laid down 
about the same things which will render these propositions 
convertible ? 

89. Into what other equivalent forms might we throw the 
joint statements that Venus is a minor planet, and minor 
planets are all large bodies revolving round the sun in 
slightly elliptic orbits within the earth s orbit ? 

90. If B is always found to coexist with A, except when 
X is Y (which it commonly, though not always, is), and if, 
even in the few cases where X is not Y, C is never found 
absent without B being absent also, can you make any 
other assertion about C ? [R.] 

91. If whenever X is present, Z is not absent, and some 
times when Y is absent, X is present, but if it cannot be 
said that the absence of X determines anything about either 
Y or Z, can anything be determined as between Z and Y ? 

[R.] 

92. If it is false that the attribute B is ever found co 
existing with A, and not less false that the attribute C is 
sometimes found absent from A, can you assert anything 
about B in terms of C ? [c.] 

93. Referring to the Elementary Lessons in Logic, p. 196, 
from the premises there given (A = AB -|- AC, B = BD, 
C = CD), derive descriptions of the terms BC, a, b, d. 

94. From the important problem of Boole, described on 
p. 197 of the same lesson, with the premises A = CD, 
BC = BD, derive descriptions of the terms BC, bC, 
B, *, d. 



xxm.] EXERCISES. 243 

95. In reference to this last named problem, examine 
each of the following assertions, and ascertain which of 
them are consistent with the premises A CD, BC = BD 

(1) ac = acD. (4) cd acd. 

(2) a = acd. (5) Kb = A^CD. 

(3) ACD = ABCD. (6) abc = abed. 

96. The premises AB = ABC, A = AB, and A = A^, 
involve self-contradiction. What is the least alteration which 
will remove this contradiction ? 

97. If AB = CD, what is the description of BD, of bd 
and 



98. What must we add to the premises, All As are Bs 
and all Bs are Cs, in order that we may establish the rela 
tion that what is not A is not C ? 

99. Verify the assertion (Principles of Science, p. 141 ; 
first edit. vol. i. p. 162) that the six following propositions 
are all of exactly the same logical meaning : 

A = BC -I- be a = bC -|- Re. 

B = AC -I- ac b = Ac -|- aC. 

C = AB -I- ab c = aR >\- Kb. 

100. Write out five similar logical equivalents of the pro 
position r PQ ! pq. 

1 01. Prove that ab = abC is equivalent to ac = aeR, 
and AB = AC to A = ABC -|- Kbc. 

102. How may the condition A [ B = ACD ! BCD 
be expressed in four non-disjunctive equations ? 

103. Verify the equivalence of M = M and N = Nm, 



244 EQUATIONAL LOGIC. [CHAP. 

when for M and N we substitute successively the following 
pairs of values : 

f M = A, f M = M, 

(I) IN = ABC. (2) \N =cD. 

, x ( M = ACD -I- AMD -I- 
(3) \ N - D -I- 



104. Express each of the following propositions equa- 
tionally in a series of non-disjunctive propositions : 

(1) Either the king is dead, or he is now on the march. 

(2) Either compression or expansion will produce either 
heat or cold in a solid body. 

(3) Kb -I- bC = G/ -\>cD. 

(4) AB -I- AC - (AB -I- AC) (G/-I- cD). 

105. In problem 20 (chap. xxi. p. 194) what description 
should we obtain of the classes c t those who do not take 
snuff, and d, those who do not use tobacco, respectively 
under the several conditions (i), (2) and (3), with (4)? 

1 06. In problem 29, pp. 200-1, draw descriptions of 
the classes At, ab y and d). 

107. Represent symbolically the logical import of the 
sentence : If it be erroneous to suppose that all certainty 
is mathematical, it is equally an error to imagine that all 
which is mathematical is certain. 

1 08. Represent equationally the logical import of this 
extract from the Oath of Supremacy : No foreign prince, 
prelate, person, state, or potentate, hath any jurisdiction, 
power, superiority, pre-eminence, or authority, ecclesiastical 
or spiritual, within this realm. Observe especially how 
far the alternatives are or are not mutually exclusive. 



xxiii.] EXERCISES. 245 

109. Take the following syllogism in Datisi : 

All men are some 1 mortals ; 

Some" men are some" 1 fools ; 
Therefore, Some iv fools are some v mortals; 
and analyse equationally the meanings of the word some 
as it occurs five times. Show which of the somes if any 
are exactly equivalent. Compare the result with the 
remarkably acute analysis of this mood given by Shedden, 
in his Elements of Logic, 1864, PP- I 3i-2. 

no. If some Xs are Ys, and for every X there is some 
thing neither Y nor Z, prove that some things are neither 
Xs nor Zs. [DE MORGAN.] 

in. Solve equationally Boole s example of analysis of 
Clarke s argument (see Laws of Thought, Chap, xiv.) 
The premises may be thus stated : 

/ ABD = 0. B/ = O. 

\ KM =0. AF - O. 

CDE = 0. A* = O. 

112. Show that every equational proposition whatsoever, 
the members of which are represented by X and Y in 
X = Y, may be decomposed into two propositions of the 
forms X = XY and Y = XY, which will not however 
always differ. Show also that the operation when repeated 
gives no new result. 

113. Take the definition Ice = Frozen Water, and throw 
it into equivalent propositions of the following forms : 

(1) One disjunctive proposition. 

(2) Two non-disjunctive propositions. 

(3) One disjunctive and one non-disjunctive. 

(4) Two disjunctives and one non-disjunctive. 

(5) One disjunctive and two non-disjunctives. 



246 EQUATIONAL LOGIC. [CHAP. 

(6) Three disjunctives. 

(7) Four non-disjunctives. 

Are these forms exhaustive, or can you frame yet other 
equivalent forms. 

114. How many and what non-disjunctive propositions 
will be equivalent to the single disjunctive, Kb >\> bC = 
Cd [ d3 ? 

115. Express the proposition AB = C ] D in the form 
of two disjunctive and then in three non-disjunctive 
propositions. 

1 1 6. As an exercise on Chapter XXII., take the 
proposition : 

Stratified Rocks = Sedimentary Rocks, and discover (i) 
one equivalent ; (2) two inferrible ; (3) several partially 
inferrible and otherwise consistent ; (4) several consistent, 
indifferent, and not inferrible ; (5) two partially inferrible, 
partially contradictory ; (6) one partially indifferent, partially 
contradictory; (7) one purely contradictory proposition. 

117. Treat in the same general manner any of the 
following premises : 

(1) Blood-vessels = arteries -|- veins. 

(2) Either thou or I or both must go with him. 

(3) Heat is conveyed either by contact or radiation. 

(4) An equation is either integrable or not integrable. 

(5) Roger Bacon, an English monk, was the greatest of 

mediaeval philosophers. 

(6) Those animals which have a brain in connection 

with a spinal cord, and they alone, have red 
blood corpuscles. [MURPHY.] 

1 1 8. Perform an exhaustive analysis of the relations of 
the following propositions, comparing each proposition 



XXIIL] EXERCISES. 247 

with each other in all the fifteen possible combinations, and 
ascertaining concerning each pair under which of the seven 
heads it falls : 

(1) A = BC. (4) a = BC abc. 

(2) Kb = Afo (5) ab = ac. 
(3) A = A; B = C (6) AB = ABC. 

1 1 9. Perform a similar exhaustive analysis of the rela 
tions of the following propositions : 

(1) Mercury = liquid, metal. 

(2) Not-mercury is not liquid. 

(3) Not-metal is either not-mercury or not-liquid. 

(4) Mercury is a metal and is liquid. 

(5) Liquid is either mercury or not-metal. 

(6) Not-liquid is either not-mercury or metal. 

(7) Not-mercury is either not-liquid or not-metal. 

The eight propositions in question 45 or the seven in 55 
of this chapter may be similarly analysed. 

120. Analyse this argument: As we can only doubt 
through consciousness, to doubt of consciousness is to 
doubt of consciousness by consciousness. 

121. Illustrate the principle that the relations of logical 
symbols are independent of space -relations. (See Prin 
ciples of Science, first ed. vol. i. pp. 39-42, 444; vol. ii. 
p. 469; new edition, pp. 32~35> 3^3, 769-) 

122. Show that if certain premises involving three terms 
leave five or more combinations unnegatived, the premises 
in question must be self-consistent 

123. From the point of view of equational logic analyse 



248 EOUATIONAL LOGIC. [CHAP. xxm. 

the metaphysical wisdom of Coleridge s doctrine of the 
syllogism thus expressed (Table Talk, vol. i. p. 207) : 

All Syllogistic Logic is i. delusion; 2. /Delusion; 
3. Conclusion - } which answer to the Understanding, the 
Experience, and the Reason. The first says : " This 
ought to be," the second adds: "This is" and the last 
pronounces : " This must be so." ; 



CHAPTER XXIV. 

THE MEASURE OF LOGICAL FORCE. 

1. THE combinational analysis of the meaning of asser 
tions enables us to establish an almost mathematical system 
of measurement of the comparative force of assertions. 
Given the number of independent terms involved, that 
form of proposition has the least possible force which 
negatives only a single combination. Thus with three 
terms, a proposition of the form AB = ABC negatives only 
the single combination ABr ; but A = ABC negatives three, 
and A = BC as many as four combinations. These latter 
propositions may be said to have three and four times the 
logical force of the first given. 

2. I have not yet been able to discover any general laws 
regarding this subject of logical force, but many curious 
and perhaps important observations may be made. Thus 
a great many forms of assertion agree in having the logical 
force one-half, that is to say, they negative half the com 
binations. Such is the case, the terms being three in 
number, with the propositions A = BC ; A = B -|- BC ; 
A = Br -|- bC. Indeed, it is very frequently true that any 
proposition having no term common to both sides of the 
equation negatives half the combinations. This is true of 
all propositions of the types A = B, A = BC, and generally 
A = BCD . . . Y. But it is not true of the type AB = CD. 



250 EQUATIONAL LOGIC. [CHAP. 

The appearance of the same term in both members of an 
equation always weakens its force ; thus A = ABC has the 
force only of f , whereas A = BC has the force i. Again, 
A = B -|. C has the force -J-, but A ! B = B -|- C only the 
force f . 

3. The best ostensive instance of logical power is found 
in a form of proposition which embraces the greatest in 
tension in one member with the greatest extension in the 
other. This kind of assertion has the general form 
ABC . . . = P .< Q -I- R !...; and as the terms increase 
the logical force approaches indefinitely to unity. Thus 
while A = B ! C has the value , AB = C ] D has that 
of 10 out of 1 6, and A -|- B = CDE that of 22 out of 32. 

A few other observations on this subject are thrown into 
the form of questions : 

4. Show that the logical force of n equations of the 

form A = B, B = C . . . . is i - 

5. Prove that a single proposition of the type ABC .... 
= P .[. Q ) R ] . . . ., there being in all n independent 
letter terms, and no term common to both members, has the 

logical force i + ^-^ which approaches indefinitely 

to unity as n increases. 

6. Can you discover any equation between a single term 
and any expression not involving that term which has a 
logical force other than one-half? 

7. What form of proposition involving only A and B in 
one member, and C, D, in the other, has the lowest possible 
logical force ? 

8. What is the utmost number of combinations of ;/ 
terms which can be negatived without producing con 
tradiction ? 



xxiv.] MEASURE OF LOGICAL FORCE. 251 

9. What is the utmost number of combinations of four 
terms which can be negatived by a proposition involving 
only three of them ? 

10. What two propositions involving five terms negative 
the utmost possible number of combinations, without self- 
contradiction ? 

11. Show that m successive propositions of the type 
A = AB, B = BC . . . ., that is to say, in the form of the 
Sorites, leave m + 2 combinations unnegatived, so that the 

, . , c . m + 2 

logical force is i - m + i 

12. Prove that the amount of surplus assertion, or over 
lapping of the propositions, in a Sorites as treated in the 
last question, increases indefinitely. Investigate the law of 
the surplusage. 

13. What is the utmost possible logical force as regards 
m terms of an equation involving n terms. 



CHAPTER XXV. 

INDUCTIVE OR INVERSE LOGICAL PROBLEMS. 

1. THE direct or deductive process of logical analysis 
consists in determining the combinations which are, under 
the Laws of Thought^ consistent with assumed conditions. 
The Inverse Problem is given certain combinations incon 
sistent with conditions, to determine those conditions. As 
explained in the Principles of Science (chapter vii.) the 
inverse problem is always tentative, and consists in invent 
ing laws, and trying whether their results agree with those 
before us. An American correspondent, Mr. M. H. 
Doolittle, points out that in making trials we should always 
pay attention to combinations in proportion to their in 
frequency , or solitariness, infrequency being the mark of 
deep correlation. The infrequency may be that either of 
presence or of absence. 

2. The following inductive problems consist of series of 
combinations of three terms and their negatives which are 
supposed to remain uncontradicted under the condition of a 
certain proposition or group of propositions. The student 
is requested to discover such propositions, express them 
equationally, and then assign them to the proper type in 
the table on p. 222. If in any problem the conditions are 
self-contradictory the student is to detect the fact. 



CHAP, xxv.] INVERSE PROBLEMS. 253 

I. IV. VII. IX. 

ABC ABC AC 

abc aEc kbc 

abC #BC 

n - aBc abc 

Kbc V. 

AC VIII. X. 

Kbc ABC ABC 



Kbc 

^BC VI. 

aRc KbC 

Kbc 
abc 

3. Assuming each of the following series of combinations 
to consist of those excluded or contradicted by certain proposi 
tions , assign the propositions which are just sufficient to 
exclude them in each problem, express these propositions 
equationally, and refer them as in the last question to the 
proper type in the table : 

I. V. VIII. 

ABC ABC ABC 



II. hbc 

abC VI abc 

III. AB IX. 

Kbc 
abc 
VII. Kbc 



abC abc 



IV. 

ABC abC 

abc abc abC 



254 



EOUATIONAL LOGIC. 



[CHAP. 



4. I now give a series of inductive problems involving 
four terms. Each series of combinations consists of those 
which remain after the exclusion of such as contradict 
certain conditions. Required those conditions. The 
problems are ranged somewhat in order of difficulty. 



I. 

ABCD 

abed 

II. 

ABCD 



III. 



abCd 
abeV 
abed 



IV. 

ABCD 



a&CD 

abed 



VI. 

ABG/ 



KbCd 
abed 



VII. 

ABCD 
ABO/ 



hbcd 
aEed 



ABC^ 


V. 


It C/t-t* 


0BCD 


ABCD 




aECd 


ABC^ 


VIII. 


^CD 


AB^ 


Kbea 


abCd 


A^CD 


aECd 


abcD 


A^C^ 


abCd 


abed 


abed 


abcft 






abed 



IX. 



aEeD 
aftea 
abCd 



X. 

ABd) 



aRcd 
abea 

XI. 

AB^D 



abCd 
abed 

XII. 

ABCD 



abed 



5. I next give a few similar problems involving five or six 
terms, as follows : 



XXV.] 



INVERSE PROBLEMS. 



255 



I. 


III. 


V. 


ABODE 


ABC^E 


AB^DE 


KBcde 


ABfl/E 


AB<;D^ 


Abcde 


ACWE 


AB^E 


aEcde 


KbcT>e 


K&cde 


abcde 


abcdE 


abCTie 




abcde 


abCde 


II. 


IV. 


VI. 


ABC/fc 


ABCDE 


ABCDE 


aECde 


KbCde 


ABC^/E 


abCde 


^BrDE 


ABd> 


abcDE 


abCDe 


A^CD^ 


abcDe 


abcdE 


^BrDE 


abcde 




abCde 



VII. 


VIII. 


ABcDef 


AECJef 


AEcdEf 


AbCVef 


aBfDef 


dBcdEE 


aBcdEf 


a&CDef 


a&CDeY 


abcDEF 



AB^DEF 



A&CVef 
AtcDEf 



IX. 

hbcDef 

AfcdEF aQcdEE 

AbcdeF ^CDEF 



a&CDef 
abcDef 
abcGEf 
dbcdeS 



256 EOUATIONAL LOGIC. [CHAP. 

6. As the reader who is in possession of the present 
volume will have plenty of unanswered inductive problems, 
it may be well to give here the answers to the problems of 
the like kind which were set in the Principles of Science^ 
new edition, p. 127. They are as follows : 

I. A = AB^- -I- AC ; a = 0BC. 
II. A = AC ; a = ciB. 

III. A = AC ; db = abc. 

IV. A = D ; B = CD -I- cd, or their equivalents 
A = D ; a = >c -I- bC. 

V. ab = abCd } Kb = 



VI. D = E ; bC = CD ; (a ! b) c = abcde. 
VII. A =^ = D =e- } B = ^B. 
VIII. (Unknown.) 
IX. btt = C -I-/; atiE = B^F ; ACF = AC^F ; 



X. This example was set by me at haphazard, like 
Nos. V. and VI. , that is to say, by merely striking out any 
combinations of the logical alphabet which fancy dictated. 
Dr. John Hopkinson, F.R.S., has given me the following 
rather complex solution 

(1) d=abd. 

(2) b = b (AF .|. ae). 

(3) A/=A/Bd)E. 

(4) E = E(B/.|-JACDF). 

(5) ^B = ^ABCDF. 

(6) abc abcef. 

(7) abef = abcef. 

Can a simpler answer be discovered ? 



xxv.] INVERSE PROBLEMS. 257 

7. In the first edition of the Principles of Science, vol. ii. 
p. 370, I gave a rather complex problem involving six 
terms, the combinations unexcluded being as follow after 
coalescence of some alternatives : 



ABCDF 

AECDef 

ABC^/E/ 

ABd)F bcd E.f 

To a request for solutions I received several answers, 
mostly correct, but, curiously enough, all differing in the 
forms of proposition. The correct ones are given below 
as furnishing a remarkable instance of logical equivalence. 

First Answer. Second Answer. 

C = ABC b = cd 

f =T)ef -\-JEf C=ABC 

c = Bd) -I- bed D = BD 

d = </E/ d = Ef 

Third Answer. Fourth Answer. 

a ac C = AC 

b = ca b = bed 

d - E/ d = E/ 

B = B (C -I- D) 

Fifth Answer. 

AC = BC 
BDE = BDEF 
B = C -I- D 
d = </E/ 

s 



258 EQUATIONAL LOGIC. [CHAP. xxv. 

The third answer, given by Mr. R. B. Hay ward, M.A.,is 
in the simplest terms. The propositions inserted as the 
fifth answer are those from which I formed the combinations 
deductively. The student may prove that any one of these 
answers is deducible from any other without descending 
explicitly to the combinations ; thus C = AC is the con- 
trapositive of a = ac ; B = C ] D is equivalent to 
b = cd t and so forth. 



CHAPTER XXVI. 

ELEMENTS OF NUMERICAL LOGIC. 

i . LET a logical term, when enclosed in brackets, acquire 
a quantitative meaning, so as to denote the number of 
individual objects which possess the qualities connoted by 
the logical term. Then (A) - number of objects possess 
ing qualities of A, or say, for the sake of brevity, the number 
of As. 

Every logical equation now gives rise to a corresponding 
numerical equation. Sameness of qualities occasions same 
ness of numbers. Hence if A = B denotes the identity of 
the qualities of A and B, we may conclude that (A) = (B). 

It is evident that exactly those objects, and those objects 
only, which are comprehended under A must be compre 
hended under B. It follows that wherever we can draw an 
equation of qualities, we can draw a similar equation of 
numbers. Thus, from A = B = C, we infer A = C ; and 
similarly from (A) = (B) = (C), meaning the number of 
As and Cs are equal to the number of Bs, we can infer 
(A) = (C). But, curiously enough, this does not apply to 
negative propositions and inequalities. For if A = B < D 
means that A is identical with B, which differs from D, it 
does not follow that 



260 NUMERICAL LOGIC. [CHAP. 

Two classes of objects may differ in qualities, and yet they 
may agree in number. 

2. The sign ] being used to stand for the disjunctive 
conjunction or, but in an unexclusive sense, it follows that 
|- is not identical in meaning with +. It does not follow 
from the statement that A is either B or C, that the number 
o As is equal to the number of Bs added to the number of 
Cs ; some objects, or possibly all, may have been counted 
twice in this addition. Thus, if we say An elector is either 
an elector for a borough, or for a county, or for a university, 
it does not follow that the total number of electors is equal 
to the number of borough, county, and university electors 
added together ; for some men will be found in two or three 
of the classes. 

This difficulty, however, is avoided with great ease ; for 
we need only develop each alternative into all its possible 
subclasses and strike out any subclass which appears more 
than once, and then convert into numbers, connected by 
the sign of addition. Thus, from A = B -|- C we get 
A = BC -I- B^ -I- BC -I- C; but striking out one of the terms 
BC as being superfluous, we have A = BC -|- B<: ] bC. 

The alternatives are now strictly exclusive, or devoid of 
any common part, so that we may draw the numerical 
equation 

(A) = (BC) + (Br) + (JQ. 
Thus, if 

A = elector, C = county elector, 

B = borough elector, D = university elector, 

we may from the proposition A = B -|- C -|- D draw 7 the 
numerical equation 

(A) = (BCD) + (BCO + (B;D) -f (B*/) + (JCD) + 

+ (bcD). 



xxvi.] QUESTIONS AND ANSWERS. 261 

3. The data of any problem in Numerical Logic will be 
of two kinds : 

(1) The logical conditions governing the combinations 
of certain qualities or classes of things, expressed 
in propositions. 

(2) The numbers of individuals in certain logical 
classes existing under those conditions. 

The qucesita of the problem will consist in determining 
the numbers of individuals in certain other logical classes 
existing under the same logical conditions, so far as such 
numbers are rendered determinable by the data. The 
usefulness of the method will, indeed, often consist in 
showing whether or not the magnitude of a class is deter 
mined or not, or in indicating what further hypotheses or 
data are required. It will appear, too, that where an exact 
result is not determinable we may yet assign limits within 
which an unknown quantity must lie. 



4. In a certain statistical investigation, among 
100 As there are found 45 Bs and 5 3 Cs ; that 
is to say, in 45 out of I oo cases where A occurs 
B also occurs, and in 5 3 cases C occurs. Sup 
pose it to be also known that wherever B is, 
C also necessarily exists. It is required to 
determine 

(1) The number of cases (all being As) where C 

exists without B. 

(2) The number of cases (all being As) where 

neither B nor C exists. 



262 NUMERICAL LOGIC. [CHAP. 

The data are as follow : 

r(A)=ioo ..... (i) 

Numerical equations -< (B) =45 ..... (2) 

l(C)= 53 ..... (3) 
Logical equation . . B = BC. 

The logical equation asserts that the class B is identical 
with the class BC, which is the true mode of asserting 
that all Bs are Cs. Two distinct results follow from this, 
namely: ist, that the number of the class BC is identical 
with the number of the class B ; and 2d, that there are 
no such things as Bs which are not Cs. 

The logical equation is thus equivalent to two additional 
numerical equations, namely, 

(B) - (BC) ....... (4) 

(B.) = o ........ (5) 

We have now means of solving the problem ; for, by the 
Law of Duality, 



By (4) 

- (B) + (JC). 
Thus 

53 = 45 + (JQ, 

or the required number of Cs is 8. 

To obtain the number of AArs, we have 

(A) = (ABC) + (ABr) + (AJC) + (Me) 
i oo =45 + +8 + (A&r). 
Hence 

(A&) = 47- 



xxvi.] QUESTIONS AND ANSWERS. 263 

5. The difference between the numbers of objects 
in any two classes whatsoever, is equal to the 
difference between the numbers of objects 
which are in each class, but excluded from the 
other class. 

Take (A) and (B) to represent the numbers in any two 
classes A and B ; then 

(A) - (B) - (AB) + (A) - (AB) - (*B) 
= (AJ) - (B). 

6. If the number of As be x, of Bs be 7, and of 
those Bs which are not As be /, then the 
number of As which are not B will be/ + x - y. 

Setting down the several logical quantities represented by 
/ 4- x - y, we have 

(*B) + (AB) + (A*) - (AB) - (*B). 
Four terms cross out, leaving only (A) as required. 



7. Represent the following argument from 
Thomson s Laws of Thought, p. 168 : 

Three-fourths of the army were Prussians ; 

Three-fourths of the army were slaughtered ; 

Therefore some who were slaughtered were 
Prussians. 

Taking A = members of the army, 
B = Prussians, 
C = slaughtered, 



264 NUMERICAL LOGIC. [CHAP. 

the premises are expressed as 

(AB) = f (A) 

(AC) = f (A). 

The number of Prussians slaughtered will be (ABC), of 
which the following equation is identically true : 

(ABC) = (AB) + (AC) - (A) + 
inserting values 
(ABC) = 



That is to say, the number of Prussians slaughtered was at 
least half the army, and exceeds it by a number equal to 
the number of men in the army who were neither Prussians 
nor slaughtered. 

8. If the number of As which are Bs is/, and the 
number of Bs which are Cs is q y what do we 
know concerning the number of As which are 
Cs? 

We have the following self-evident equations : 

AC - ABC + AC 

= ABC + AB;- + ABC + tfBC -f 

+ dEc - B 
= AB + BC - B + AC + 



Inserting the values given, we get 

AC =/ + q - B + AC + 

We see that the data are quite insufficient for determining 
the number of As which are Cs. They may be anything 
from zero up to the whole number of As or Cs. To make 
the question determinate we need also the number of Bs, 



xxvi.] QUESTIONS AND ANSWERS. 265 

as well as the number of ACs, which are not Bs, and the 
number of Bs which are neither A nor C. 



9- Most Bs are As ..... (i) 

Most Bs are Cs ..... (2) 

Therefore, Some Cs are As ..... (3) 

The above argument is a celebrated one, proposed by 
De Morgan (Formal Logic, p. 163), and discussed by Boole 
(Trans, of the Cambridge Philosophical Society, vol. xi. part 
ii. p. i) and others. Regarded as an ordinary Aristotelian 
pseudo-syllogism, it is subject to the fallacy of undistributed 
middle, since the proposition Most Bs are As/ must be 
counted as a particular affirmative. The pseudo-mood is 
accordingly 1 1 1 in the third figure. Nevertheless the force 
of the argument is pretty obvious and may thus be analysed. 

The mark of quantity, most, of course means more than 
half, and is one of the few quantitative expressions used in 
ordinary language. We can easily represent the two pre 
mises in the form 



- ....... (i) 

(BC) = i(B) + / ....... (2) 

To deduce the conclusion, we must add these equations 
together, thus, 

(AB) + (BC) - (B) + w + w . 

Developing the logical terms on each side, we have 

(ABC) + (ABc) + (ABC) + (^BC) - (ABC) + (AB*) 
+ (0BC) + (B<r) 4- w + w . 

Subtracting the common terms, there remains 
(ABC) = w + w + (aBc). 



266 NUMERICAL LOGIC. [CHAP. 

The meaning is, that there must be some Cs which are 
As, amounting to at least the sum of the quantities w and 
w , the unknown excesses beyond half the Bs which are As 
and Cs. The number (aBc) is wholly undetermined by the 
premises, but it cannot be negative. Thus w + w is the 
lower limit of (ABC). 



IO. (i) For every Z there is an X which is Y ; 

(2) Some Zs are not Ys. What inferences can 
be drawn ? 

This general problem given by De Morgan in his Syllabus 
(p. 29, art. 85), and in some other parts of his writings, 
would thus be represented in my formulas, which differ 
essentially, however, from those of De Morgan. 

The premises are 

(XY)-(Z) + i (i) 

in which m would be zero if De Morgan meant that there 
are not more Xs which are Ys than there are Zs, but just an 
equal number 

(Zy) = (2) 

where n is some positive number. 
Developing (i) we get 

(3) (XYZ) + (XYz) - (XYZ) + (XyZ) + (xYZ) 
+ (xyZ) + ;//. 

Striking out the common term and adding (Xyz) to both 
sides, we have for the number of Xs which are not Zs 

(Xz) = m +n+ (Xyz) + (xYZ). 



xxvi.] QUESTIONS AND ANSWERS. 267 

Again, after striking out the common term, equation (3) 

reduces to 

(XYz) = (XyZ) + (xZ) + m, 

which gives as the number of Zs which are not Xs 
(xZ) = (XYz) - (XyZ) - ;;/. 

The student should compare these results with those of the 
less general problem given in the Principles of Science, new 
edition, p. 169 ; first edition, vol. i. p. 191, and also with 
De Morgan s results expressed in a totally different kind of 
notation. 

II. If m or more Xs are Ys, and n or more Ys 
are Zs, what do we know about the number of 
Xs which are therefore Zs ? 

This question represents one case of the numerically 
definite syllogism as treated by De Morgan (Syllabus, p. 27). 
Taking X, Y, and Z to be the three terms of the syllogism, 
he adopts the following notation : 

u - whole number of individuals in the universe of 

the problem. 
x = number of Xs. 
y = number of Ys. 
z number of Zs. 

Making m denote any positive number, wXY means, in 
De Morgan s system, that m or more Xs are Ys. Similarly 
;?YZ means that n or more Ys are Zs. Smaller Roman 
letters denote the negatives of the larger ones. Thus wXy 
means that m or more Xs are not Ys, and so on. 

From the two premises wXY and YZ, De Morgan draws 
the conclusion (m + n - jy)XZ. Let us consider what 



268 NUMERICAL LOGIC. [CHAP. 

results are given by our notation. The premises may be 
represented by the equations 

(XY) = m + m (YZ) = n + , 

where m and n are the same quantities as in De Morgan s 
system, and m and n two unknown but positive quantities, 
indicating that the number of XYs is m or more, arid the 
number of YZs is n or more. 

The possible combinations of the three terms X, Y, Z, and 
their negatives are eight in number, and these all together 
constitute the universe, of which the number is u. The 
problem is at once seen to be indeterminate in reality ; for 
there are eight classes of which the numbers have to be 
determined, and there are only six known quantities, namely, 
z/, x, y, z, m, and , by which to determine them. Accord 
ingly we find that De Morgan s conclusion, though not 
absolutely erroneous, has little or no meaning. From the 
premises he infers that (m + n - y) or more Xs are Zs. 
Now 

m + n-y= (XY) + (YZ) - (Y) - ; - n 
= (XYZ) - (xYz) - m - n . 

Thus De Morgan represents the number of the whole class, 
XZ, by a quantity indefinitely less than its own part, XYZ. 
It is quite true that if the second side (XYZ) - (xYz) 

- m - n of this equation has value, there must be at 
least this number of Xs which are Zs ; but as (xYz) 
may exceed (XYZ) in any degree, this may give zero or a 
negative result, while there is really a large number of XZs. 
The true and complete expression for the number of XZs 
is found as follows : 

(XZ) = (XYZ) + (XyZ) 

- (XYZ) + (XYz) + (XYZ) + (xYZ) - (Y) + (XyZ) 
+ (xYz) = /// + m + n + n - y + (XyZ) + (xYz). 



xxvi.] QUESTIONS AND ANSWERS. 269 

Among these seven quantities, only ;, ;z, and y are definitely 
known. The two m and ri are two indefinite quantities, 
expressing the uncertainty in the number of XYs and 
YZs, while there are two other unknown quantities, the 
numbers of XyZs and xYzs arising in the solution of the 
problem. 

12. If m or more Xs are Ys, and n or more Ys 
are Zs, what do we know about the number of 
not-Xs which are not-Zs ? 

From the same two premises as in the last problem, 
namely 

mXY and YZ, 

De Morgan draws the conclusion 

(m + n 4- u - x - y - z)xz 

that is to say, the number of not-Xs which are not-Zs is 
the quantity in the brackets or more. This conclusion is 
equivalent to that in the preceding problem. 

To prove this result it is requisite to develop all the 
combinations numbered in each of the quantities ;//, n t u, 
x, y, z ; there are twenty-six terms in all which the reader 
may readily work out. Giving them the signs indicated by 
De Morgan, and striking out pairs of positive and negative 
terms, we find only two combinations left, together with 
m and n t which terms are used, as in the last problem, to 
express the fact that De Morgan s proposition ;;zXY is not 
really definite, but means that m or more, that is m or 
(/ + m } Xs are Ys. We thus obtain 

(xy) - (xyz) - (XyZ) - m - n 

in which the term (XyZ) is wholly undetermined. Thus we 
find that De Morgan s method gives us as the value of (xy) 



270 NUMERICAL LOGIC. [CHAP. 

a part of itself (xyz), diminished by three unknown quanti 
ties. The number (xz) may accordingly be of any magni 
tude, while the lower limit assigned to it by De Morgan is 
zero, or even negative. The problem is in fact a wholly 
indeterminate one, and De Morgan s solution is illusory. 

Similar remarks may be made concerning other conclu 
sions which De Morgan draws. Thus, from ;;/Xy and nYz 
(mXs or more are not Ys, and ;*Ys or more are Zs) he infers 

(m + n x) xZ and (m -\-n-z) Xz. 

But it will be found by analysis that the first of these results 
has the following meaning : 

(xZ) ^ (xYZ - (XYz) ; 

that is to say, the lower limit of the class xZ is a part of 
itself, xYZ, diminished by the number of another class 
XYz of unknown magnitude. 

13. If the fractions a and fi of the Ys be severally 
As and Bs, and if a + {3 be greater than unity, 
it follows that some As are Bs. 

[Cambridge Phil. Trans, vol. x. part i. p. 8.] 

In his third memoir on the Syllogism De Morgan gives 
the above as a very general statement of the conditions of 
valid mediate inference. He remarks that the logician, 
that is to say, the ordinary Aristotelian logician, demands 
a = i or ft = i, or both; he can then infer. This 
represents the condition of a distributed middle term. 

The numerically definite conditions are readily represented 
as follow : 

The premises are a . (Y) = (AY). 

ft . (Y) = (BY). 



xxvi.] QUESTIONS AND ANSWERS. 271 

Hence 

(a + /?)()= (AY) + (BY) 



= (ABY) + (AY) + (ABY) + (*BY), 
(a + p) (Y) - (Y) - (ABY) - (abV), 
or 

(ABY) = ( a + j8 - i) (Y) + (ab\). 

We learn that the number of AYs which are Bs is the 
fraction ( + ft - i ) of the Ys, together with the undeter 
mined number (0Y), which cannot be negative. But, 
according to the conditions, a + (3 is greater than unity ; 
hence the second side of the equation must have a positive 
value. Not only will there be (a + /3 - i) As which are 
Bs, but this is merely the lowest limit, and there will be 
as many more as there are units in the number of 0Ys. 

If we distribute the middle term Y once, by making 
a = i , we have 

(ABY) = /? . (Y) + o. 

The term (abY) of course vanishes because the whole of 
the Ys are As. Again, if /3 = i ? we have 

(ABY) = a . (Y). 

If both a and /3 become unity, then 
(ABY) - (Y). 

It must be carefully noted however that these results do not 
show the whole number of As which are Bs, but only those 
which are so within the sphere of the term Y. Nothing 
has been said about the combinations of not-Y, which are 
quite unlimited by the conditions of the problem. 

14. If A occurs in a larger proportion of the 
cases where B is than of the cases where B is 



272 NUMERICAL LOGIC. [CHAP. 

not, then will B also occur in a larger propor 
tion of the cases where A is than of the cases 
where A is not. 

This general proposition is asserted in J. S. Mill s chapter 
On Chance and its Elimination, but is not proved by 
Mill. (System of Logic, Book III., chapter xvii. section 2, 
adfinem; fifth edition, vol. ii. p. 54.) I do not remember 
seeing any proof of it given elsewhere, and it is not to my 
mind self-evident. The following, however, is a proof of its 
truth, and is the shortest proof I have been able to find. 

The condition of the problem may be expressed in the 
inequality 

(AB) : (B) > (A*) : (b\ 

or reciprocally in the inequality 

(B) : (AB) < (*) : (M). 
Subtracting unity from each side, and simplifying, we have 

(0B) : (AB) < (ab) : (Alt). 

Multiplying each side of this inequality by (Ab) : (B) we 
obtain 

(Ab) : (AB) < (ab) : (aB). 

Restoring unity to each side, and simplifying 

(A 

or reciprocally 



which expresses the result to be proved, namely, that B 
occurs in a larger proportion of the cases where A is than 
of the cases where A is not. 

15. In a company of r individuals, p have coats 
and q have waistcoats. Determine some other 
relations between them. 



xxvi.] QUESTIONS AND ANSWERS. 273 

Boole treats this problem in the fourth page of his 
Memoir On Propositions numerically definite (Cambridge 
Philosophical Transactions, vol. xi. part ii.). Taking i to 
represent the company which is the universe of the propo 
sition, x the class possessing coats, y the class possessing 
waistcoats, and using the letter N, according to Boole s 
notation, as equivalent to the words number of, 

p = NX, q = Ny, r=Ni, 
he finds, as we have found in a preceding page (p. 2 6 4, No. 7) 



Nxy = p + $ - r + Ni x i y. 



Ni x i y = r p q + Nxy. 

He proceeds, * Again, let us form the equation 

2 P ~ $ ~ r = 2 N# - Ny - N i . 
= N (2x - y - i) 



= N (xi y 2yi x i xi -y) 



= NX i - y - 2Ny i-x-Ni-xi-y. 
From which we have 



NX i y 2p q r-\- 2Ny i - x + N i x i y. 

Hence we might deduce that the number who had coats 
but not waistcoats would exceed the number 2p q r 
by twice the number who had waistcoats without coats 
together with the number who had neither coats nor waist 
coats. This is not, indeed, the simplest result with reference 
to the class in question, but it is a correct one. 

The student is requested to verify this result. 

On going over this paper of Boole s again, it becomes 
apparent to my mind that his method is identical with 
that developed in this chapter and in my previous paper 

T 



274 NUMERICAL LQGIC. [CHAP. 

on the same subject (Memoirs of the Manchester Literary 
and Philosophical Society, Third Series, vol. iv. p. 330, 
Session, 186970), written with a knowledge, as stated on 
p. 331, of Boole s publication on the subject. 



16. Can we represent a syllogism in the extensive 
form by means of numerical symbols ? 

In a very interesting and remarkable paper read to the 
Belfast Philosophical Society in 1875, Mr. Joseph John 
Murphy has given a kind of numerical notation for the 
syllogism. He has since printed a more condensed and 
matured account of his views in Mind, January, 1877. 

Taking the syllogism Chlorine is one of the class of 
imperfect gases ; imperfect gases are part of the class of 
substances freely soluble in water ; therefore, chlorine is 
one of the class of substances freely soluble in water he 
assumes the symbols 

x Chlorine, y = Imperfect gases, 

z = substances freely soluble in water. 
He expresses the first premise in the form 

y*x +P, 

p being a positive numerical quantity indicating that there 
are other things besides chlorine in the class of imperfect 
gases. The second premise takes the form 
z=y + f, 

similarly indicating that besides imperfect gases there are 

q things in the class of substances freely soluble in water. 

Substitution gives z = x +p + q, 

which would seem to prove that besides chlorine (x) there 

are p + q things in the class of substances freely soluble in 

water. 



xxvi.] QUESTIONS AND ANSWERS. 275 

The student who wishes to master the difficulties of the 
modern logical views should read these papers with great care. 
Space does not admit of my arguing the matter out at full 
length, and I can therefore only briefly express my objec 
tions to Mr. Murphy s views as follows : His equations are 
equations in extension, and, with his use of + and - , they 
can only hold true when his terms are numerical quantities. 
Under this assumption his equations show with perfect 
correctness the numbers of certain classes; but they are not 
therefore equivalent to syllogisms. Because z = x + p + q, 
we learn that the number z exceeds x by p + #, but it does 
not therefore follow that chlorine belongs to the class of 
substances represented by z. In short, as I have pointed 
out at the beginning of this chapter (p. 259), from logical 
equations arithmetical ones follow, but not vice versa. (See 
also Principles of Science, p. 171 ; first edit. vol. i. p. 193.) 
I hold, therefore, that Mr. Murphy s forms are not really 
representations of syllogisms ; but at the same time I am 
quite willing to admit that this is a question never yet 
settled and demanding further investigation. It is very 
remarkable that Hallam inserted in his History of Literature 
(ed. 1839, vol. iii. pp. 287-8) a long note containing a 
theory of the syllogism somewhat similar to that of Mr. 
Murphy, but which has hitherto remained unknown to Mr. 
Murphy and apparently to all other logical writers. 



CHAPTER XXVII. 

PROBLEMS IN NUMERICAL LOGIC. 

1. IF from the number of members of Parliament we 
subtract the number of them who are not military men, we 
get the same result as if from the whole number of military 
men we subtract the number of them who are not members 
of Parliament. Prove this. 

2. In a company of x individuals it is discovered that y 
are Cambridge men, and z are lawyers. Find an expression 
for the number of Cambridge men in the company who are 
lawyers, and assign its greatest and least possible values. 

[BOOLE.] 

3. Prove that in any population the difference between 
the number of females and the number of minors is equal 
to the difference between the number of females who are 
not minors, and of minors who are not females. 

4. Show that if to the number of metals which are red, 
we add the number which are brittle, the sum is equal to 
that of the whole number of metals after addition of the 
number of metals which are both red and brittle, and after 
subtraction of the number of metals which are neither red 
nor brittle. 

5. What is the value of the following expression 

(A)-(AB)-(AC)? 



CHAP, xxvii.] PROBLEMS. 277 

6. Prove that the number of quadrupeds in the world 
added to the number of beings not quadrupeds which 
possess stomachs is equal to the whole number of things 
having stomachs together with the number of things not 
having stomachs which are quadrupeds. 

7. If x and y be respectively the numbers of things which 
are X and Y, while m is the whole number which are both 
X and Y, and n the number which are either X alone 
or Y alone, what is the relation between m + n and 
x+y? 

8. Let u be the whole number of things under con 
sideration, x the number which are A, and y the number 
which are B ; then if m be the number of things which are 
both A and B, show that m + u x y is the number 
which are neither A nor B. 

9. Taking each logical term to represent the number of 
things included in its class, verify the following equa 
tions : 

(A - AB) (A - AC) = A - AB - AC + ABC - Kbc. 

(A - AB) (A - AC) (A - AD) = A - AB - AC + 

ABC - AD + ABD + ACD - ABCD = Kbcd, 

10. What is the product of the logical multiplication of 
the four factors 

(A - AB) (A - AC) (A - AD) (A - AE)? 
Give another expression for its value. 

11. Show that the following equation is necessarily true : 

B + AC + C + A&: = A + C + aEc. 



12. What happens in Problem 8 if it be discovered that 
the class B does not exist at all ? 

13. Find an expression for the difference between (A) 
and (B) + (C). 



278 NUMERICAL LOGIC. [CHAP. 

1 4. What is (a) the minimum percentage of C that must, 
and (/?) the maximum that may coincide with B under the 
following conditions ? 

80 per cent of As coincide with 50 per cent of Bs. 

70 per cent of As coincide with 60 per cent of Cs. [D.] 

15. If revolutions occur in a larger proportion of govern 
ments where the press is under a censorship, than of govern 
ments where it is not, then will a censorship of the press be 
found in a larger proportion of governments which are sub 
ject to frequent revolutions, than of governments which are 
not thus subject. [D.] 

1 6. If/ per cent of A are B, and q per cent of A are 
C, what is the least percentage of A that those individuals 
make up which are both B and C ? [D.] 

1 7. Show that we cannot tell what percentage of B or of 
C the same individuals make up unless we know how much 
of B or of C is not A. [D.] 

1 8. In the easy case in which all B is A, and all C is A, 
find what percentage of B or of C must be made up by the 
individuals which are both B and C at once. [D.] 

1 9. Prove the following equations : 
(Afe) + (AB) = (A) + (ABC) - (AC). 

(A + B) - (C + D) = (A + B) (c + C<t) - (C + D) 
(a + A6) - AB (Cd + cD) + CD (A + 0B). 

20. Prove that the following equation gives a correct 
expression for the common part of any three classes - 
A, B, C. 

(ABC) = (B) + (C) - (A) - (aB) - (aC) + (Ate). 

21. In a company consisting of r individuals there were 
q in number who knew Latin, and / in number who knew 



xxvn.] PROBLEMS. 279 

either Latin or French, but not both ; between what limits 
is the number of those who knew French confined ? 

22. In the last problem prove that the lower limit is the 
greatest value in / - q and q - p, and the upper limit, the 
least value in 2r - p - q, and/ + q. (See Boole, On Pro 
positions Numerically Definite, p. 15.) 

23. The student will find many other numerically defi 
nite problems in De Morgan s Formal Logic, Chapter VIII., 
and in his Syllabus, pp. 27-30; but in reading De Morgan 
it must be carefully remembered that mXY means with him 
not that mXs are Ys but that m or more Xs are Ys. His 
solutions will sometimes, as shown in the previous chapter, 
be found delusive. 

24. Verify the following assertion of De Morgan: To 
say that mXs are not any one to be found among any lot of 
72 Ys is a spurious (that is a self-evident or necessary) pro 
position, unless m + n be greater than both x and y, in 
which case it is merely equivalent to both of the following, 
(/// + n y] Xy, and (m + n - x) Yx, which are equiva 
lent to each other. 

25. It is found that there are in a certain club of x mem 
bers, y London graduates, and z lawyers. What further 
numerical data are requisite in order to define the numbers 
who are both London graduates and lawyers, and of those 
who are neither ? 

26. If there are more persons in a town than there are 
hairs on any one person s head, then there must be at least 
two persons in the town with the same number of hairs on 
their heads. Put this theorem into a strict logico-mathe- 
matical form. [HERBERT SPENCER.] 

27. Demonstrate the theorem in numerical logic given in 
the Principles of Science, new edition (only), p. 170. 

28. For every man in the house there is a person who 



280 NUMERICAL LOGIC. [CHAP. xxvn. 

is aged; some of the men are not aged; it follows, and 
easily, that some persons in the house are not men ; but 
not by any common form of syllogism. (DE MORGAN, 
Syllabus, p. 29.) A solution of this problem is given in 
Principles of Science, new edition, p. 169. 

29. Draw what conclusions you can from the following : 
There were some English on board; and though no 
passengers were saved from the wreck, and of the ship- 
officers, as it happened, only one, yet no Englishman was 
lost. [R.] 



CHAPTER XXVIII. 

THE LOGICAL INDEX. 

1. I NOW give what I propose to call the Logical Index, 
or, more precisely, the Logical Index of Three Terms. As 
however the logical relations of two terms are too simple 
to need an index, and those of four terms are vastly too 
numerous and complex to admit of exhaustive treatment 
at present, the Index of Three Terms is practically the only 
one which can be given. It contains, within the space of 
four pages, a complete enumeration of all possible purely 
logical conditions involving only three distinct terms. 

2. Each page contains a double-sided table, forming in 
fact two tables. Each such table contains a column of 
equational propositions, a column of Roman numerals 
showing the type (see p. 222) to which such propositions 
belong, a column of consecutive Arabic numbers for sake 
of easy reference, and lastly a column of Greek letters, 
which supplement the Greek letters a, /?, y, given at the 
heads of the columns of propositions. These Greek letters 
stand in place of the combinations of the fourth column of 
the Logical Alphabet (p. 181), as follow : 

a = ABC e = 0BC 

/? - AB<: = aEc 

y = AC rj = abC 

8 = Kbc 9=abc 



282 THE LOGICAL INDEX. [CHAP. 

It is obvious that each Greek letter appearing in the middle 
column of the Index represents the presence of the corre 
sponding combination, or rather its non-exclusion. Absence 
of the Greek letter represents exclusion. Looking, for 
instance, to No. 31, we learn that a = bc, an assertion of 
the Vlth type, excludes all combinations except a, ft y, 
specified at the top of the table, and 0, specified in the 
centre column ; that is to say, the combinations consistent 
with a = bc are ABC, ABr, AC, and abc. The principal 
use of the Index, however, will be in the inverse direction, 
to find the law corresponding to certain unexcluded com 
binations. Taking, for instance, the combinations A&r, #BC, 
a&C, their Greek signs are 5, e, r; ; to find their law, then, 
we must look in the last table in the column headed (not-a), 
(not-/?), (not-y), and in the line showing 8, c, 17, in the 
middle column. We there find the two assertions A = c 
- Kb of the IVth type (No. 230), as those corresponding 
to the combinations in question. 

3. With the aid of this Index we can infallibly and 
rapidly solve all possible problems relating to three terms. 
What assertion, for example, can we make which shall not 
be contradictory to, and yet shall not be inferrible from, the 
premise a = BC -\~abct Working out the combinations 
unexcluded by this premise, we find them to be ABr, AC, 
A&r, # BC, and abc, or /3, 7, S, e, 0. Of these, /? and 8 may be 
removed simultaneously without wholly removing any letter, 
that is to say without contradiction. Looking in the second 
table of the Logical Index at No. 8 1, we find the proposition 
A = AC (of type II.) as one which removes /3 and 8. 
This is the required proposition which is, as it were, quite 
neutral to the one assumed. In the same way we might 
remove 7 and without contradiction, so that No. 34, or 
Ab = be, of type XII., is another neutral proposition. It 



xxviii.] THE LOGICAL INDEX. 283 

may, I believe, be safely inferred that every proposition of 
type XIII., that of the premise in question, will have at 
least two propositions neutral to it, of types II. and XII. 
respectively. 

Suppose it be required, as a second instance, to define the 
precise points of agreement and difference of two disput 
ants, one of whom asserts that ( i ) Space = three-way 
spread with points as elements (Henrici) ; while his oppo 
nent holds that (2) Space = three-way spread, and at 
the same time (3) Space has points as elements, but is not 
known to be the only thing that has. The three assertions 
are symbolised as below, the combinations excluded being 
indicated by their Greek signs : 

f/3 

(,) A-Bdjf < 2 t = Ar." 

(3) A = AC V 

u 

We see that the second disputant s assertions have a 
logical force superior to that of the first by J, namely , 
which corresponds to assertion 5, or a~B = a~BC. In addition, 
then, to all that the first asserts, he affirms that a three-way 
spread which is not space has points as elements. 

As a third instance of the power and flexibility of this 
combinational logic, suppose it to be required to make an 
exhaustive statement of all the inferences which can be 
drawn from the theorem that Similiar figures (A) consist 
of all whose corresponding angles are equal (B), and whose 
corresponding sides are proportional (C). We proceed in 
this way. The proposition is of the form A = BC, of the 
Vlth type, and negatives ft y, 8, e. Any proposition, then, 
which negatives one, two, or three of these combinations 



284 THE LOGICAL INDEX. [CHAP. 

will be inferrible from the theorem, but not equivalent to it. 
All the possible inferences, therefore, are indicated in the 
following table of Index Numbers, which, taken in con 
nection with the Logical Index, sufficiently explains itself: 

ft 65 /3 y 97 Py8 113 

7 33 S 81 /3ye 105 

8 17 ft 73 ft 8 e 89 

e 9 y 8 49 y 8 6 57 

7 e 4i 



These fourteen assertions, which are all the possible 
non-equivalent inferences, or the equivalents of these, were 
detected by the Logical Index in a few minutes ; it would 
be doubtfully possible, and in any case a most laborious 
problem, to obtain an exhaustive statement of inferences 
by any other method, if indeed any other method exists. 

The want of space alone prevents my giving more abun 
dant illustrations of the multitudes of logical problems 
which may be solved infallibly and speedily by the use of 
the Logical Index. It may be safely said that in four pages 
of tables it gives the key to all possible logical questions, 
relations or problems involving three distinct logical terms. 
There is some possibility that the corresponding index 
for the relations of four terms may some day be worked out, 
as, when exhibited in like manner, it will occupy only one 
volume of 1024 pages of a rather larger size than those of 
this volume. There is no prospect whatever that the corre 
sponding index for five terms will ever be exhaustively 
published, since it would fill a library of 65,536 volumes, 
each containing 1024 large pages. This fact will give some 
faint idea of the possible number and complexity of logical 
relations involving only a very moderate number of terms. 



THE LOGICAL STAMP. 285 



The Logical Stamp. 

In my previous logical books x I described a Logical 
Slate with five series of the combinations of the Logical 
Alphabet engraved upon it. I first made such a slate in 
May 1863, an d I have since frequently used it with much 
saving of labour. The recent extensive introduction of 
india-rubber printing stamps lately suggested to me the 
idea that the most convenient method of obtaining the 
logical combinations would be to stamp them on paper. 
Two stamps producing the combinations of three and of 
four terms as shown in columns IV. and V. of the Logical 
Alphabet (p. 181), were made for me at a cost of about 
eleven shillings. 

They have been very successful, and leave nothing to -be 
desired as regards the private study of logical problems. 
One great advantage of the stamps over the slate is evident, 
namely, that the work being done on paper can be preserved 
for reference without copying. 

The ABCD stamp can readily be utilised for problems of 
five, six, or more terms. For six terms, for instance, it is 
requisite to make four impressions and distinguish them by 
writing EF, E/j eF, ef, above the respective impressions. 

India-rubber stamps of any design can now be easily 
ordered at all of the principal stationers. 

1 Pure Logic, 1864, p. 68 ; Substitution of Similars, 1869, p. 54 ; 
Elementary Lessons in Logic, 1870, p. 199 ; Principles of Science, 1874, 
Vol. I., p. 1 10 ; New Editions, p. 96. 



286 



THE LOGICAL INDEX. 



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XXVIII.] 



THE LOGICAL INDEX. 



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XXVIII.] 



THE LOGICAL INDEX. 



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CHAPTER XXIX. 

MISCELLANEOUS QUESTIONS AND PROBLEMS. 

IT seems convenient to bring these Studies in Deductive 
Logic to a close by adding a certain number of mixed 
Questions and Problems, which may refer to any part of 
logical doctrine. In some cases these questions pass the 
bounds of formal and deductive logic. It is left to the 
student to determine what part, if any, of the preceding 
pages will assist him. To certain questions, however, are 
appended references to other works where the proper 
assistance will be found. 

1. What may we expect to happen, in a logical point of 
view, when an irresistible force meets with an infinite 
resistance ? 

2. If it is said to be false that what has the properties 
A and B has those also of C and D, and vice versa, how 
would you interpret this statement as affecting the possible 
relations of A, B, C, and D ? 

3. In how many ways may we in a purely logical point of 
view contradict the assertion of Hobbes that Irresistible 
might in the state of nature is right ? (See p. 182, 
Question 7.) Specify the ways. 

4. Analyse the logical import of the following passage 
from the Wealth of Nations, book i. chapter viii. : 



CHAP, xxix.] MISCELLANEOUS QUESTIONS. 291 

It is not because one man keeps a coach, while his 
neighbour walks afoot, that the one is rich, and the other 
poor ; but because the one is rich, he keeps a coach, and 
because the other is poor, he walks afoot. 

5. In Harriet Martineau s Autobiography (vol. i. p. 355) 
we are told that a certain lady, after receiving from Charles 
Babbage a long explanation of his celebrated calculating 
machine, terminated the conversation with the following 
question : < Now, Mr. Babbage, there is only one thing more 
that I want to know. If you put the question in wrong, 
will the answer come out right? 

If you think this question absurd, give distinct and 
detailed reasons for thinking so, and reconcile them with 
the fact that false premises may give a true conclusion. 

6. Explain and illustrate the Aristotelian saying: Ex 
veris fieri non potest ut falsum condudatur ; ex falsis contra 
verum; and show some of its applications in the investiga 
tion of nature. r R 1 

7. A certain argument having been shown to involve 
paralogism, inquire into the conditions under which this 
failure does or does not tend to establish the contradictory 
conclusion. 

8. In a certain borough, on one occasion, the Liberal 
party objected to 3624 voters, and the Conservative party 
to 553 1 voters, the whole constituency being 10,000. 
What is the least number of voters which can have been 
objected to on both sides ? What is the greatest number ? 
What is the most probable number, supposing the objections 
to be made quite at haphazard ? 

9. What is the logical, compared with the popular, in 
terpretation of the injunction, This man is not on any 
account to be ducked in the horse-pond. Explain the 
difference. 



292 MISCELLANEOUS QUESTIONS [CHAP. 

10. What is the logical, compared with the popular, in 
terpretation of the injunction, All persons are requested 
not to discharge fireworks among the crowd around the 
bonfire on the 5th November. 

11. Because a horse is an animal, the head of a horse 
is the head of an animal. 

Examine the validity of this inference. Can you express 
the reasoning syllogistically, or symbolically ? [E.] 

12. Investigate the nature of the reasoning, good or bad, 
involved in the four following examples : 

(1) Elephants are stronger than horses; horses are stronger 

than men; therefore elephants are stronger than 
men. [E.] 

(2) Alexander was the son of Philip ; therefore Philip 

was the father of Alexander. 

(3) As good kill a man as kill a good book ; for he that 

kills a man does but kill a reasonable creature ; 
but he that kills a good book kills Reason herself. 

(4) Nay, look you, I know tis true ; for his father built a 

chimney in my father s house, and the bricks are 
alive at this day to testify it. [o.] 

13. What methods underlie these inferences? 
Because it froze last night, therefore the pools are 

covered with ice. 

During the retreat of the Ten Thousand a cutting north- 
wind blew in the faces of the soldiers ; sacrifices were offered 
to Boreas, and the severity of the wind immediately ceased, 
which seemed a proof of the god s causation. [p.] 

1 4. What method is employed in the following ? 

* Brewster accidentally took an impression from a piece 
of mother-of-pearl in a cement of resin and bees wax, and 
finding the colours repeated upon the surface of the wax, 



xxix.] . AND PROBLEMS. 293 

proceeded to take other impressions in balsam, fusible 
metal, lead, gum-arabic, etc., and always found the iri 
descent colours the same. He thus proved that the chemical 
nature is wholly a matter of indifference, and the form of 
the surface is the condition of such colours. 

15. What is the difference, logically, between the sen 
tences : Leibnitz, a great philosopher, has said, etc. ; and 
A great philosopher, Leibnitz, has said, etc. ? [c.] 

1 6. What is successive induction, or induction by con 
nection, as in the proof that ?z 2 = 1 + 3 + 54- 

up to n of the odd numbers ? [H.] 

(See Elementary Lessons in Logic, lesson xxvi. p. 220.) 

1 7. Give an inductive proof that x n a n is divisible by 
x a, when n is a whole number. 

1 8. Can there be such a thing as a fallacy of simple in 
spection, that is a fallacy which does not involve inference ? 
(See Mill s System of Logic, book iv. chapter iii.) 

1 9. Leibnitz says Knowledge is either obscure or clear. 
The clear is again either confused or distinct : and the 
distinct either adequate or inadequate; is further either 
symbolical or intuitive ; and if it be at the same time both 
adequate and intuitive, it is perfect. 

Give an exhaustive classification of all possible kinds of 
knowledge under the above conditions. (See Elementary 
Lessons in Logic, lesson vii.) 

20. Explain the logical meanings of the terms Genus, 
Species, Difference, Property, and Accident, distinguishing the 
meaning in extent and intent, and using for illustration the 
varieties of things ABCD, ABG/, ABd), AB^, in which 
A, B, C, D are terms denoting qualities, and c, d the 
negatives of C, D. 

21. Draw the inferences deducible from these data: 



294 MISCELLANEOUS QUESTIONS [CHAP. 

(1) A is the only antecedent always present when p is 
present, and always absent when / is absent. 

(2) A is an antecedent always present when p is present, 
and always absent when p is absent. 

(3) A is an antecedent frequently present when / is 
present, and frequently absent when/ is absent. [p.] 

22. Point out the exact nature of the relations between 
the logical processes of Abstraction, Analysis, Synthesis, 
and Generalisation. 

23. What is the logical difference, if any, between nouns 
Substantive and nouns Adjective ? 

24. Is a Latin adjective used alone in the neuter an 
adjective or a substantive ? 

25. Is there self-contradiction in the assertion that know 
ledge of what is outside my consciousness may be inside 
my consciousness ? 

26. Can absolute certainty be found in any conclusion (i) 
inductively established, (2) deductively established? [E.] 

27. Is there any distinction, and if so what, between a 
general and an abstract notion, and is there a corresponding 
difference between the names employed to express them ? 

[E.] 

28. It is a rule of syllogism that nothing can be inferred 
from particular premises. How then can I infer from the 
particular facts that some men have died, the universal 
conclusion, All men die ? [E.] 

29. Explain the limits of demonstrative science, and 
examine the following statement : No matter of fact can 
be matter of demonstration. [E.] 

30. Distinguish Logical, Mathematical, and Physical 
Quantity. [ K ] 

31. Distinguish and exemplify Logical, Mathematical, 
Metaphysical, Physical, and Moral Necessity. [E.] 



xxix.] AND PROBLEMS. 295 

32. When two phenomena are causally connected to 
gether, can you always ascertain which is the cause and 
which is the effect ? If so, how ? [L.] 

33. Investigate how far or on what grounds our know 
ledge of the following propositions approximates to 
certainty : 

* Nitric acid does not dissolve gold. 

A distant fixed star is subject to gravity. 

34. Consider from a logical point of view the assertion 
that the increasing trade of Great Britain is caused by a 
reform of the tariff. What kind of proof is applicable ? 

35. A man having been shot through the heart imme 
diately falls dead. Investigate the logical value of such a 
fact as proving that all men shot through the heart will fall 
dead. [L] 

36. What do you understand by a working hypothesis ? 
Under what conditions is it legitimate for an investigator to 
employ hypothesis ? (Huxley. Mill s System of Logic, book 
iii. chap. xiv. ; Principles of Science, chap, xxiii.) [L.] 

37. State these arguments formally, and give their tech 
nical designations : 

(1) The thinking power does not belong to matter; 

otherwise matter generally would exhibit it. 

(2) Happiness is the reward of goodness ; and since all 

do not desire a good life, all cannot obtain its 
reward. [p.] 

38. Why is it that with exactly the same amount of 
evidence, both negative and positive, we did not reject the 
assertion that there are black swans, while we should refuse 
credence to any testimony which asserted that there were 
men wearing their heads underneath their shoulders ? [P.] 



296 MISCELLANEOUS QUESTIONS [CHAP. 

39. What is the difference of meaning, if any, between 
the propositions, This house was built by Jack, and 
This is the house that Jack built ? (De Morgan, Third 
Memoir on the Syllogism, loth page.) 

40. Does the thesis that the ultimate premises in human 
knowledge are the result of mental association affect the 
nature and certainty of Logic, and if so, how ? [E.] 

41. Define evidence. Distinguish intuitive, demonstra 
tive, and probable evidence. [E.] 

42. Explain : Certainty, therefore, has for its opposite, 
uncertainty in one way impossibility in another. Uncer 
tainty, in the language of logicians, is its contradictory 
opposite impossibility, its contrary opposite. [P.] 

43. Investigate the question whether the truth of a 
statement is to be judged by the impression which it makes 
upon those to whom it is addressed, by its literal corre 
spondence with the belief of the person making it, or by any 
other standard. [L.] 

44. It has been pointed out by Ohm that reasoning to 
the following effect occurs in some works on mathematics : 
A magnitude required for the solution of a problem must 
satisfy a particular equation, and as the magnitude x 
satisfies this equation, it is therefore the magnitude required. 
Examine the logical validity of this argument. 

45. It is probable that Herodotus recorded only what he 
heard concerning Ethiopia ; and it is not unlikely that most 
that he heard was correct; so that we may accept his 
account as true. Is this conclusion correct ? 

46. There is a very strong probability that the eldest 
child of a newly married couple will inherit the estate of 
the husband. For, firstly, it is more probable than not that 
there will be children of the marriage. Next, if a child is 
born, it is more probable that it will be a son, for more boys 



xxix.] AND PROBLEMS. 297 

are born than girls. Thirdly, if a son be born, it will 
probably survive its father. Examine this inference, [o.] 

47. Consider the following argument : Many writings 
that are not genuine were ascribed to Clemens Romanus ; 
this Epistle was ascribed to him ; therefore this Epistle is 
not genuine. [L.] 

48. A student of geometry examines three isosceles 
triangles and finds them agree in having equal angles at the 
base; an excise officer examines three bottles of wine out 
of a quantity imported and finds them agree in strength ; 
a chemist analyses three specimens of a mineral and finds 
them agree in composition : compare the inferences which 
may be drawn in these cases. 

49. What is the relation between classification and in 
duction in general ? [L.] 

50. When an experiment designed to produce a phe 
nomenon fails to produce it, in how many ways may we 
interpret or explain the meaning of the failure ? [L.] 

51. In what ways may a physicist hope to explain away 
an exceptional phenomenon ? [L.] 

52. If we never find skins except as the teguments of 
animals, we may safely conclude that animals cannot exist 
without skins. If colour cannot exist by itself (airav yap 
Xpw/xa ev o-GjpxTi), it follows that neither can anything that 
is coloured exist without colour. So, if language without 
thought is unreal, thought without language must also be so. 
What do you think of this argument ? [o.] 

53. If we are disposed to credit all that is told us, we 
must believe in the existence not only of one, but of two or 
three Napoleon Buonapartes ; if we admit nothing but what 
is well authenticated, we shall be compelled to doubt the 
existence of any. How, then, can we be called upon to 
believe in the one Napoleon Buonaparte of history? [o.] 



298 MISCELLANEOUS QUESTIONS [CHAP. 

54. Brown asserts that all planets are spheroids; Jones 
denies it ; Robinson asserts that Jones knows nothing about 
the matter; Smith proves that in this case at least Robinson 
is correct ; but Thomson refuses to accept the premises of 
Smith s proof. What are the logical relations of the 
parties ? 

55. From the statement that blood-vessels are either 
veins or arteries, does it follow logically that a blood-vessel, 
if it be a vein, is not an artery ? Give your reasons. 

5 6. It is asserted by some philosophers that all knowledge 
is inductive in its origin, and it is generally allowed that 
inductive inferences can be probable only ; if so, no know 
ledge can be more than probably true. Can you, however, 
adduce any instance of knowledge which is certainly true ? 
In that case explain the difficulty which evidently arises. 

57. Aut amat aut odit mulier ; nihil tertinm. If any 
one takes upon himself simply to deny the truth of this 
saying of Publius Syrus, in how many different ways may 
the denial be interpreted ? 

5 8. Explain the following apparent paradox : P thinks 
of an object ; Q is absolutely ignorant of the size of that 
object ; to him, therefore, the probability that the object is 
greater than a cannon-ball is J. Again, being absolutely 
ignorant about its size, he has no reason to believe it either 
greater or less than a pea, the probability of either case 
being -J. Hence to him it is infinitely improbable that 
the object is intermediate in size between a pea and a 
cannon-ball. [JOHN HOPKINSON, D.SC] 

59. In defending a prisoner his counsel must either deny 
that the deed committed is a crime, or he must deny that 
the prisoner committed the deed ; therefore if the counsel 
denies that the deed committed is a crime, he must admit 
that the prisoner did commit the deed. 



xxix.] AND PROBLEMS. 299 

60. What do you understand by the logical proof of an 
assertion ? Compare the logical meaning of the word proof 
with any other meanings of the word known to you. [i.] 

6 1. Can all kinds of propositions be exhibited in the 
intensive as well as the extensive form ? Give reasons in 
support of your answer ; and in the event of its being in 
the negative, draw up a list distinguishing between those 
kinds of propositions which can, and those which cannot, 
be so exhibited. [L.] 

62. Explain the meaning of the assertion that Induction 
is the inverse process of Deduction. 

63. Illustrate Mathematical Induction in its several kinds 
or cases, and discuss its relation to induction in the physical 
sciences. 

64. What is the relation, if any, between the inductive 
syllogism and the inductive methods employed in the 
physical sciences ? 

65. Estimate upon logical grounds the possibility of 
establishing a school in which students should be rendered 
capable of discovering the Laws of Nature. (Gore s Art of 
Scientific Discovery^) 

66. What precisely is meant by the Law of Continuity? 
Point out the grounds and limits of its validity. (Life of 
Sir W. Hamilton, p. 439 ; Principles of Science, chapter 
xxvii.) 

67. When the effects of three distinct causes are added 
and mingled together, by what processes of experiment and 
reasoning can we assign to each cause its separate effect ? 

[a] 

68. Under what circumstances are we to accept the 
failure of an experiment or series of experiments as proving 
the non-existence of the phenomenon intended to be 
produced? (Principles of Science, chapter xix.) [L.] 



300 MISCELLANEOUS QUESTIONS [CHAP. 

69. Illustrate the scientific value of exceptional pheno 
mena, and show in how many ways they may be disposed 
of or reconciled with physical law. (Principles of Science, 
chapter xxix.) [L.] 

70. What is the difference between the causal and the 
casual happening of events, if, as is generally allowed, not 
even a dead leaf falls to the ground without sufficient causes 
to determine the precise moment of its falling and the 
precise spot upon which it will fall ? 

71. Show by example that the logical copula does not 
imply the notion of existence. [E.] 

72. Investigate the question whether the functions of 
affirmative and negative propositions in reasoning are 
similar. 

73. England is the richest country in the world, and has 
a gold currency. Russia and India, in proportion to 
population, are poor countries, and have little or no gold 
currency. How far are such kinds of facts logically 
sufficient to prove that a gold currency is the cause of a 
nation s wealth? [i.] 

74. If by two distinct methods of investigation you arrive 
at the same conclusion, namely, that the currency of the 
kingdom does not exceed one hundred millions sterling, but 
it is afterwards discovered that one of the methods of 
investigation involved fallacious reasoning, what would you 
be inclined to infer about the other method of investigation? 

M 

75. A certain argument having been shown to involve 
paralogism, inquire into the conditions under which this 
failure does or does not tend to establish the contradictory 
conclusion. 

76. Investigate the logical, psychological, and moral 
grounds of the saying, Qui s excuse, s accuse? 



xxix.] AND PROBLEMS. 301 

7 7. Taking the senses in which they most resemble one 
another, distinguish between judgment, opinion, statement, 
knowledge, fancy, conjecture, supposition, allegation. [E.] 

78. Distinguish: truth, certainty, fact, opinion, proba 
bility, evidence, conviction. [E.] 

79. How far can the inconceivability of the opposite be 
regarded as proof of the truth of any judgment ? [E.] 

80. Right-angled and not-right-angled are contradictory 
predicates; therefore, according to the law of Excluded 
Middle, as the proposition All triangles are right-angled 
is false, it must be true that all triangles are not right- 
angled. But this also is false. Explain the above difficulty. 

[E.] 

8 1. Given that (i) whenever the statements a, b, x are 
either all three true, or all three false, then the statement 
c is false, and y is true, or else c is true, and y is false ; 
(2) that whenever d, e, y are either all three true or all three 
false, then the statement a is false, and x is true, or a is true, 
and x is false. When can we infer from these premises that 
either x or y is true ? 

[Hugh MacColl, B.A., in Educational Times, question 
6206. A solution was given by C. J. Monro, M.A., 
in the same paper for March 1880. The question 
seems to mean What other conditions with those 
given determine that either x or y is true ?] 

82. De Morgan says (Fourth Memoir on the Syllogism, 
p. 5) of the Laws of Thought : Every transgression of 
these laws is an invalid inference ; every valid inference is 
not a transgression of these laws. But I cannot admit that 
everything which is not a transgression of these laws is a 
valid inference. Investigate the logical relations between 
these three assertions. 



302 MISCELLANEOUS QUESTIONS [CHAP. 

83. To what type of assertion do the premises of Darapti 
belong ? 

84. Give the converse, inverse, contrapositive, obverse, 
and reciprocal propositions of the following : 

(1) All parallelograms have their opposite angles equal. 

(2) If P is greater than Q, then R will be greater 

than S. 

(3) Two triangles are congruent if the three sides of the 

one are respectively equal to the three sides of the 
other. 

85. Why have some mathematicians been accustomed to 
say that it is necessary to prove the converse of a mathematical 
proposition ? 

86. Where exactly lies the error of the Irishman, who 
being charged with theft on the evidence of three witnesses 
who had seen him stealing the article in question, proposed 
to bring in his defence thirty witnesses who had not seen 
him stealing it ? 

87. Epimenides says that every statement of a Cretan is 
a lie ; but Epimenides is a Cretan ; therefore what he says 
is a lie ; therefore every statement of a Cretan is not a lie. 

[E.] 

88. If in saying that few strikes are beneficial, I feel 
sure that the statement will be misinterpreted by those to 
whom it is addressed, and that the statement no strikes 
are beneficial, although not in my opinion literally true, 
will more exactly convey to the hearers minds the im 
pression which I believe to be true, ought I, having regard 
to the moral obligation of speaking the truth, to use the 
latter assertion or the former ? 

89. I will go on/ said King James, I have been only 
too indulgent. Indulgence ruined my father. 



xxix.] AND PROBLEMS. 303 

Express clearly the process of reasoning involved in this 
utterance. Is it Induction ? or what ? [M.] 

90. What is the relation, if any, between the inductive 
syllogism and the inductive methods employed in the 
physical sciences ? 

91. Can the proposition, All A is all BJ be regarded as 
representing a single act of thought ? (See Mind, vol. i. 
P- 216.) [ K ] 

92. Are the premises of Darapti given only in a 
numerical form sufficient to prove the conclusion ? 

93. Does it follow that, because some poetry is not in 
verse, there must be some verse which is not poetry ? [H.] 

94. Take the proposition, All sciences are useful, and 
determine precisely what it affirms, what it denies, and what 
it leaves doubtful, concerning the relations of the terms 
science and useful thing. 

95. Ascertain precisely how many distinct assertions 
there are in the description of the conduct of the great 
scholastic logician, John of Salisbury, after Thomas a 
Becket had been murdered by his side : Tacitus sed mcerens, 
continue se subduxit. 

96. Can you represent equationally the contradiction 
between Some Xs are not some Fs and There is one X 
only and that is the only F ? 

9 7. Which of the types of assertion involving three terms 
are complete, in the sense of admitting no additional assertion 
involving the same three terms without self-contradiction ? 

98. If all things are either X or F, and all things are 
either F or Z, what inference can you draw ? 

99. Do the thirty-six moods of Hamilton s Syllogism with 
quantified predicates (see table in Elementary Lessons, 
p. 1 88 ; Thomson s Laws of Thought, section 103) comprise 
all the possible weakened moods ? 



304 MISCELLANEOUS QUESTIONS. [CHAP. xxix. 

100. Is the student of logic, generally speaking, prepared 
rapidly to analyse the two following propositions, and to say 
whether or no they must be identical, if the identity of 
synonyms be granted ? 

(1) The suspicion of a nation is easily excited, as well 

against its more civilised as against its more war 
like neighbours, and such suspicion is with 
difficulty removed. 

(2) When we see a nation either backward to suspect 

its neighbour, or apt to be satisfied by expla 
nations, we may rely upon it that the neighbour is 
neither the more civilised nor the more warlike of 
the two. 

[DE MORGAN, Third Memoir ; 1858, p. 181.] 

1 01. Is the following proposition a definition or not ? Is 
it on the matter or the form of the proposition that you 
found your answer? 

LOGICA EST ARS ARTIUM ET SCIENTIA SCIENTIARUM. 



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